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Title: Sturm-Liouville Equations with Singular Endpoints
of Poincaré Rank Zero and One
by
Jeremy L. Mandelkern
A dissertation submitted to the College of Sciences at
Florida Institute of Technology
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in
Applied Mathematics
Melbourne, Florida
May, 2017
We the undersigned committee hereby approve the attached dissertation,
“Sturm-Liouville Equations with Singular Endpoints of Poincaré Rank Zero and One”,
by Jeremy L. Mandelkern.
_________________________________________________
Advisor: Dr. Charles Fulton
Professor
Mathematical Sciences
_________________________________________________
Committee Member: Dr. Gnana Tenali
Professor
Mathematical Sciences
_________________________________________________
Committee Member: Dr. Kanishka Perera
Professor
Mathematical Sciences
_________________________________________________
Committee Member: Dr. Ivica Kostanic
Associate Professor
Electrical and Computer Engineering
_________________________________________________
Department Head: Dr. Ugur Abdulla
Professor and Head of Mathematics
Mathematical Sciences
iii
ABSTRACT:
Sturm-Liouville Equations with Singular Endpoints
of Poincaré Rank Zero and One
Jeremy Mandelkern
Advisor: Dr. Charles Fulton
In this dissertation, my objective is to study, for (0, ),x the equations
(1) 2( ) 0,xy x x y of Bessel, and (2) 41 0,y x y with a view to
obtaining representations of solutions near the singular endpoints, connection formulas
for solutions defined near one singular endpoint in terms of solutions defined at the other
singular endpoint, and to make use of such information to prepare the way for spectral
theoretic investigations involving the Titchmarsh-Weyl m function and spectral density
function for equation (2). The work is motivated in part by the formulation of the
“Sturm-Liouville Connection Problem” recently formulated by Charles Fulton in [47]
(See p. 158.). In all of the differential equations investigated in this dissertation, it has
been the classification of the singular endpoints that has played the central role in the
methods that have been developed. The basic definition of Poincaré rank is as follows:
For the 2nd order equation ( ) ( ) ( ) ( ) ( ) 0, (0, ),w x f x w x g x w x x where ( )f x
and ( )g x are rational functions for which either ( )f x or ( )g x or both ( )f x and ( )g x
have poles at 0 [0, ).x x Then the singular point 0x x has Poincaré rank 1
where is the least integer for which both 0( ) ( )x x f x and 2
0( ) ( )x x g x contain no
iv
powers of 0x x in their denominators [83, p.148], [10]-[15]. To determine the Poincaré
rank of a singularity at ,x one may transfer this singularity to 0t by means of the
inversion substitution 1x t and apply the scheme described above for the resulting t
equation. The rank of 0t is equal to the rank of .x A rank of 0 is the classical
case of a regular singular point, and a rank of 1 is the case of an irregular singular point.
Below is a summary of the content contained within each chapter of this dissertation.
Chapter 1 addresses equations with regular singular endpoints whereby a new matrix
method of solution is given with an application shown towards the Bessel equation of
non-integral order.
In Chapter 2 we investigate equation (2), which has irregular singular points of rank 1
[21] at both 0x and .x Here a new method is given to formulate principal and
canonical non-principal solutions near the LP irregular singular endpoint 0,x which are
entire in . A similar approach is taken to make definitions of solutions near x for
Im 0. Existence and uniqueness of the solutions is established. The Sturm-Liouville
Connection problem is solved using the solutions defined at 0x and .x
Chapter 3 addresses a new characterization of the spectral density function ( )f
given by Fulton, Pearson, and Pruess [45]. This method makes use of the Appell System,
a companion linear system of ordinary differential equations. Here the spectral density
function for a case involving the Bessel equation is obtained demonstrating the first
nontrivial example of a spectral density function calculation using this new technique.
v
Table of Contents
Table of Contents .............................................................................................................. v
List of Tables and Figures ............................................................................................... vii
Acknowledgement .......................................................................................................... viii
Dedication ......................................................................................................................... ix
Chapter 1: Equations with Regular Singular Points ......................................................... 1
A Matrix Formulation of Power Series Solutions ............................................................. 1
Chapter 1 Introduction ...................................................................................................... 1
Section 1: Matrix Forms of Frobenius Theory for Second Order Equations .................... 2
Section 2: Preliminaries for the Vectorization Procedure; The Bessel Equation.............. 5
Section 3: A Vectorization Procedure ............................................................................... 9
Section 4: Connection to Bessel Functions ..................................................................... 14
Section 5: An Example of The Vectorization Procedure; Theorem 2............................. 16
Section 6: A Vectorization Procedure for the General Problem ..................................... 21
Section 7: Some History on the Bessel Equation and the Power Series Method ........... 24
Summary of Results ......................................................................................................... 28
Chapter 2: Sturm-Liouville Equations with Irregular Singular Points ........................... 29
The 41 x Potential; Irregular Singular Points at 0x and x ....................................29
Chapter 2 Introduction .................................................................................................... 29
Section 8: The 41 x Potential SL Equation and Irregular Singular Point 0x ........... 30
Section 9: The 41 x Potential SL Equation and Irregular Singular Point x ........... 60
vi
Existence of a Bounded Solution to the Terminal Value Problem .................................. 60
Section 10: Uniqueness of the Bounded Solution ........................................................... 69
Section 11: Asymptotics For The Solutions Defined Near x .................................. 81
Section 12: The Connection Problem Preliminaries ....................................................... 85
Section 13: A Related Investigation of W. Bühring’s .................................................... 99
Section 14: The Connection Problem ........................................................................... 116
Section 15: Determination of the Characteristic Exponents ......................................... 159
Section 16: Some Brief Considerations on the 1 x Potential .................................... 162
Summary of Results ....................................................................................................... 165
Chapter 3: New Methods in Spectral Theory .............................................................. 166
A New Calculation of the Spectral Density Function for Bessel’s Equation ................ 166
Chapter 3 Introduction ................................................................................................... 166
Section 17: Background: The Titchmarsh-Weyl m-Function ...................................... 166
Section 18: The Appell System ..................................................................................... 168
Section 19: Calculation of the SDF for Bessel’s Equation ........................................... 173
Section 20: Validation of Results ................................................................................. 178
Summary of Results ....................................................................................................... 179
Conclusion ..................................................................................................................... 179
References ...................................................................................................................... 180
vii
List of Tables & Figures
Table 1 – The Bessel Equation, Power Series Method, and Matrix Formulations .......... 25
Figure 1 – Suitable Contours I ....................................................................................... 111
Figure 2 – Related Equations, Transformations, and Parameter Restrictions ............... 121
Figure 3 – Suitable Contours II ...................................................................................... 122
viii
Acknowledgement
Special thanks to my PhD advisor Dr. Charles Fulton of the Florida Institute of
Technology for his unwavering support and guidance, without which,
none of this would have been possible. Also, in particular, I’m grateful to him for his
significant contributions made in Section 8, the content of which was joint work and is
used in this dissertation with his permission. I also reserve deep gratitude for my advisor,
W. Balser, and W. Bühring for the great inspiration that I found in their works.
ix
Dedication
I dedicate this dissertation to my late Father and to my Mother. They did their very best
as parents to instill in me a strong work ethic, integrity, and many qualities that have
enabled me to continue on in the face of hardship when things were looking most bleak.
I also dedicate this dissertation to my lovely wife, Li Wen Fang. Without her continued
encouragement and support, I could have never reached the finish line.
1
Chapter 1: Equations with Regular Singular Endpoints
A Matrix Formulation of Power Series Solutions
Chapter 1 Introduction: In Coddington and Levison [25, p. 119, Thm. 4.1] and Balser
[9, p. 18-19, Thm. 5], matrix formulations of Frobenius theory, near a regular singular
point, are given using 22 matrix recurrence relations yielding fundamental matrices
consisting of two linearly independent solutions together with their quasi-derivatives. In
this chapter we apply a reformulation of these matrix methods to the Bessel equation of
nonintegral order and a generalization of the method is then given in Section 6. The
reformulated approach of this paper differs from [25] and [9] by its implementation of a
new “vectorization” procedure that yields recurrence relations of an altogether different
form; namely, it replaces the implicit 22 matrix recurrence relations of both [25] and [9]
by explicit 44 matrix recurrence relations that are implemented by means only of 44
matrix products and sums. From this reformulated approach, a matrix recursion
involving products of 44 matrices emerges by which the power series representations
of two Bessel functions and their quasi-derivatives are obtained. This approach appears
to be new; see, for example, the historical observations in Section 7. The new idea from
this chapter of using a vectorization procedure may further enable the development of
symbolic manipulator programs for matrix forms of the Frobenius theory.
2
Section 1: Matrix Forms of Frobenius Theory for Second Order Equations
Consider the 1st order system with ( )x being a 22 matrix,
(1.1) 1
( ) ,X x Xx
where 0
( ) n
n
n
x A x
is convergent for x a for some 0.a The system (1.1) is said
to have a regular singular point (RSP) at 0x and will therefore have two linearly
independent solutions defined for ,0 0, .x a a We now give the matrix
formulation of Frobenius theory from [25, p. 119, Thm. 4.1, eq. 4.4] and [9, p. 20, ex. 2]
stated as Theorem 1.
Theorem 1: Assume 0A has two distinct real eigenvalues 1r and 2 ,r 1 2 ,r r 1 2 ,r r n
1,2,3,...n . Let P be the invertible change of basis matrix for which
(1.2) 11
0 0
2
0.
0
rB P A P
r
Under the above assumptions, every system (1.1) has a unique fundamental matrix
solution of the form
(1.3) 0( ) ( ) ,B
Y x T x x 0
0
( ) , ,n
n
n
T x T x T P
0 ,x a
where the 22 matrix coefficients nT are uniquely determined by the recurrence relation
(1.4) 1
0 0
0
( ) ,n
n n n m m
m
T B nI A T A T
1.n
3
Remark 1: When ( ) ( ) ( ) 0P x y Q x y R x y with a RSP at 0x is converted to the
system form (1.1), the indicial roots, 1r and 2r , of this scalar equation, are identically the
eigenvalues of 0.A
Remark 2: 0B in (1.2)-(1.4) is a “monodromy matrix” (See [68, p.439] or [97, p. 485].)
and may be calculated as in (1.2) by a diagonalization of 0A while 0Bx may then be
explicitly computed by means of the formula 0
0exp ln .B
x B x
Remark 3: The “uniqueness” of the 22 matrix solution in (1.3) results from the
initialization 0T P and from [9, p. 212, Lemma 24] or [50, p. 225, eq. 32] which will be
discussed further in section 2.
We now state a definition from Balser [9, p. 18].
Definition 1: System (1.1) has a “good spectrum” if no two eigenvalues of 0A differ by
a natural number.
Using the definition above, we now give the matrix formulation of Frobenius theory from
[25, p. 119, Thm. 4.1, eq. 4.2] and [9, p. 19, Thm. 5] stated as Theorem 2.
4
Theorem 2: Every system (1) with good spectrum has the unique fundamental matrix
solution
(1.5) 0( ) ( ) ,A
X x S x x 0
( ) ,n
n
n
S x S x
0 ,S I 0 ,x
where the 22 matrix coefficients nS are uniquely determined by the recurrence relation
(1.6) 1
0 0
0
( ) ,n
n n n m m
m
S A nI A S A S
1n .
Remark 4: Any implementation of recursion relations (1.4) or (1.6) poses considerable
difficulty as these relations prohibit explicit representations of the unknown matrices nS
or nT in terms of previous 'nS s or 'nT s owing to their appearance once on the left and
once on the right in matrix products. To overcome this difficulty indicated above, we
now introduce the new idea in the next section, a vectorization procedure.
5
Section 2: Preliminaries for the Vectorization Procedure; The Bessel Equation
The first appearance of a Bessel function was in a 1738 memoir by Daniel Bernoulli [18],
[104, p. 356] and since then, numerous books and papers have been devoted to their
properties ([104, ch XVII], [101], [32, Ch. XIX], [33, Ch. VII], [35], [89, Ch. 1,4],
[1, Ch. 9-11], [38], [40]-[41], [19], [27], [52]-[53], [56], [73], [81], [84], [89]-[91]).
For the Bessel Equation,
(2.1) 2
( ) 0, 0,xy x y xx
we obtain the 1st order system (1.1) as demonstrated by Baker in [8], where 1.
Thus we set
(2.2) y
Xxy
to yield
(2.3) 1
( )X A x Xx
where
(2.4) 2
0 1 2( )A x A A x A x with 0 2
0 1,
0A
12 2
,x
A and 2
0 0
0A
.
Before we apply Theorem 1 and later Theorem 2 to system (2.3)-(2.4), we first observe
that the assumptions 1 2 , 1,2,3,r r n n , from Theorem 1 and that of a good
spectrum from Theorem 2 prohibit both the nonzero integer and the nonzero half-integer
6
values of the indicial roots of (2.1), 1,2 .r However, for system (2.3)-(2.4), the
assumptions of Theorems 1 and 2 may be slightly relaxed to admit the nonzero half-
integer values of . This analysis follows from a theorem of Kaplan [69, p. 369, Thm. 5,
eq. 9-40]. Recall in the scalar Frobenius theory, with the solution ansatz 0
,n r
n
n
y c x
(2.1) gives rise to the two-term recurrence relation
2 ,n nc cn r n r
hence the theorem of Kaplan applies ensuring that two linearly independent non-
logarithmic power series solutions exist whenever the indicial roots 1,2r of (2.1) satisfy
1 2r rn
k
, 1,2,3,...n For the Bessel equation (2.1), Kaplan’s parameter k is 2 , so it
follows that the fundamental matrix solutions to (2.3)-(2.4), given by both Theorems 1
and 2, are valid so long as 1,2,3,... Summarizing, the assumptions on the indicial
roots 1,2r , in Theorems 1 and 2, are sufficient (but not necessary) to exclude all the
logarithmic cases in the Frobenius theory as required for Theorems 1 and 2 to apply. Our
objective now is to apply the matrix form of Frobenius theory, as stated in Theorem 1, to
obtain a fundamental matrix solution ( , )Y x to the system form of the Bessel equation
(2.3)-(2.4). To this end, we compute the monodromy matrix 0B in (1.2) by a
diagonalization of 0A whose eigenvalues are 1,2 .r We find
(2.5) 1
0 0
0,
0B P A P
7
where the matrices 1
1 12 2
,1 1
2 2
P
0 2
0 1,
0A
and 1 1
.P
Next we compute the matrix exponential 0Bx as
(2.6)
0
0
0
0
0 01exp ln ln
0 0!
1ln 0
! 0.
1 00 ln
!
n
B n
n
n n
n
n n
n
x x xn
xn x
xx
n
Observe now that the recurrence relation (1.4) for the Bessel equation (2.3)-(2.4)
becomes for 2,3,4n ,
(2.7) 22
0 0 1 0 0,
0 0 0n n n
nT T T
n
with the initialization given by (1.3) as 0
1 1.T P
It follows from 1
2 2,
xA that
12 2x
T and then from (2.7) that
(2.8) 2 1
2 2n
xT , 1,2,3,... .n
Now to get 2T from 0 ,T we observe that for 2,n (2.7) becomes
(2.9) 2 2 2 2
2
2 2 2 2
2 0 0 1 0 0,
0 2 0
a b a b
c d c d
where we have set 2 2
2
2 2
.a b
Tc d
Here we can now observe the main obstacle in solving
8
(2.9) for 2 :T namely, the matrix 2T appears in a “commutator” form, multiplying a 22
matrix once on the left and once on the right, thus prohibiting the explicit solution for 2T
in the matrix form of the relation (2.9). However, 2T is uniquely determined by matrix
theory from Gantmacher [50, p. 225, eq. 32] or the restatement as a lemma in Balser [9,
p. 212, eq. A.1], which asserts that the general commutator matrix equation for square
matrices ,A ,B ,C and ,X
(2.10) ,A X X B C
has a unique solution X if A and B have no common eigenvalues. This criteria holds
in (2.9) with
(2.11) 2
0 1,
0A
2 0,
0 2B
for 1,2,3,...,
so the existence of a unique solution of (2.9) for 2X T follows directly from this matrix
theory.
9
Section 3: A Vectorization Procedure
To overcome the commutator equation (2.9), so that a solution procedure for 2T and more
generally nT emerges, we now introduce a rather simple approach that seems to have
eluded others: namely, we will first write the four scalar equations for the four unknowns
2 2 2, , ,a b c and 2d (well-known) and then generate, by means of products of suitable
44 matrices (the new idea), an explicit solution formula. This first yields, by
computation of the matrix products in (2.9), the equivalent 44 linear system
(3.1)
2 2
2 2
2
2 2
2
2 2
2 0
2 0
2
2
a c
b d
c a
d b
.
Now we introduce a “vectorization” of the matrices 0 2 4, , ,T T T . as follows:
(3.2)
0
0
0
0
0
1
1:
a
bT
c
d
where 0 0
0
0 0
1 1,
a bT P
c d
and for 1,2,3,...,n
(3.3)
2
2
2
2
2
:
n
n
n
n
n
a
bT
c
d
where 2 2
2
2 2
n n
n
n n
a bT
c d
.
Thus the rows of the 22 matrices 2nT are “stacked” on top of one another to form the
41 vectors 2 .nT Now in the second stage of our vectorization procedure, the 44 linear
10
system (3.1) is re-written as
(3.4) 2 2 0 ,M T T
2 2
2
2 0 1 0
0 2 0 1,
0 2 0
0 0 2
M
and where
0 0 0 0
0 0 0 0: .
0 0 0
0 0 0
By the assumption that 1, 2M is invertible, so
(3.5)
1
2 2 0 2
2
2 10 0
4 1 4 1
2 1 0 0 0 0 10 0
4 1 4 1 0 0 0 0 1
2 0 0 00 0
4 1 4 1 0 0 0
20 0
4 1 4 1
T M T
2
2
2
2
2 1
2 1.
2 1
2 1
11
Finally, we revert from the vector form of the solution to obtain the 22 matrix solution
to (2.9) as
(3.6)
2 2
2 2
2
2 2
2 2
2 1 2 1.
2 1 2 1
a bT
c d
Clearly, the 44 matrix equation (3.4) is preferable to the 22 commutator matrix
equation (2.9). Accordingly, we can now proceed to generate a “closed form” solution of
the matrix recurrence relation (2.7) for the Bessel equation by employing the above
vectorization procedure in the general case. Thus, putting 2n j in (2.7) and taking
2 2
2
2 2
,j j
j
j j
a bT
c d
2
2
2
2
2
,
j
j
j
j
j
a
bT
c
d
then the recurrence relation (2.7) becomes
(3.7) 2 2 2 2j j jM T T for 1,2,3,...j ,
where
(3.8)
2 2
2
2 0 1 0
0 2 0 1.
0 2 0
0 0 2
j
j
jM
j
j
By the assumption that 1, 2, 3, , 2 jM is invertible and hence:
(3.9) 1
2 2 2 2 ,j j jT M T
12
where computation from Mathematica gives
(3.10) 1
2 2
2
2 10 0
4 ( ) 4 ( )
2 10 0
4 ( ) 4 ( ).
20 0
4 ( ) 4 ( )
20 0
4 ( ) 4 ( )
j
j
j j j j
j
j j j jM
j
j j j j
j
j j j j
Now proceeding to implement the recursion for 2 2, 2 4, , 2n j j gives
(3.11) 1 1 1 1 1
2 2 2 2 2 2 2 2 4 2 0( )( )( ) ( ) ,j j j j j jT M T M M M M T
1,2,3,...j .
So, by our vectorization procedure, it follows that (2.7) becomes:
(3.12) 1
1
2 2 2 0
0
,j
j j N
N
T M T
1,2,3,4,5,...j .
As all matrices in (3.12) are known, we can compute the product, obtaining
(3.13)
2
2
2
2
2
( 1)
2 ! ( 1)
( 1)
2 ! 1
,
( 1) 2
2 ! ( 1)
( 1) 2
2 ! (1 )
j j
j
j
j j
j
j
jj j
j
j
j j
j
j
j v
j v
T
j v
j v
j
j
where ( ) ( )( 1) ( 1)j j is the Pochhammer symbol.
13
Finally, we revert from vector form to get the closed form 22 matrix solution of (2.7) as
(3.14)
2 2
2
2 2
( 1) ( 1)
2 ! ( 1) 2 ! 1
.
( 1) 2 ( 1) 2
2 ! ( 1) 2 ! (1 )
j j j j
j j
j j
jj j j j
j j
j j
j v j v
T
j v j
j v j
From (1.3), (2.6), and (3.14), the fund. system to (2.3)-(2.4) given by Theorem 1 is then
(3.15)
01 2 2
2
01 2
, , 0( , ) ( )
, , 0
B j
j
j
y x y x xY x T x x T x
xy x xy x x
2 2
2 20 0
2
22 20
2 20 0
( 1) ( 1)
2 ! ( 1) 2 ! 10
.0 ( 1) 2 ( 1) 2
2 ! ( 1) 2 ! (1 )
j j j j j j
j jj jj j
j
jj j j j j jj
j jj jj j
x x
j v j vx
T xx j v x j x
j v j
As system (2.3)-(2.4) is the matrix formulation of the Bessel Equation (2.1), we may
expect the fundamental matrix (3.15) to contain Bessel Functions. This connection will
now be made in the next section.
14
Section 4: Connection to Bessel Functions
From (3.15), the Bessel functions and the modified Bessel functions are not yet
transparent. This is because these special functions receive their definitions from the
standard Frobenius theory, which uses a scalar recurrence relation for the equations:
(4.1) 2 2 2( ) 0,x u xu x u Bessel Equation
(4.2) 2 2 2( ) 0,x u xu x u Modified Bessel Equation
where no eigen-parameter appears. The Bessel functions then receive their definitions
by making special choices of constants multiplying the Frobenius solutions
( )iu x , 1,2,3,4,i to (4.1) and (4.2), where 1 1( ) ( ,1),u x y x 2 2( ) ( ,1),u x y x
3 1( ) ( , 1),u x y x and 4 2( ) ( , 1),u x y x so that
(4.3)
2
1
0
1 ( 1)( ) ( ) ,
2 ( 1) ! ( 1 ) 2
jj
v
j
xJ x u x
j j
(4.4)
2
2
0
1 ( 1)( ) ( ) ,
2 ( 1) ! (1 ) 2
jj
v
j
xJ x u x
j j
(4.5)
2
3
0
1 1( ) ( ) ,
2 ( 1) ! ( 1 ) 2
j
v
j
xI x u x
j j
(4.6)
2
4
0
1 1( ) ( ) .
2 ( 1) ! (1 ) 2
j
v
j
xI x u x
j j
We now wish to identify which Bessel functions are generated in the fundamental matrix
solution ( , )Y x in (3.15). Comparison between (3.15), (4.3)-(4.6) reveals that the entries
15
of ( , )Y x contain weighted Bessel functions of the arguments x and x or more
specifically,
(4.7)
/2 /2
/2 /2
/2 /2
/2 /2
2 ( 1) ( ) 2 (1 ) ( )
2 ( 1) ( ) 2 (1 ) ( )
( , )2 ( 1) ( ) 2 (1 ) ( )
2 ( 1) ( ) 2 (1 ) ( )
v v
v v
v v
v v
J x J x
d dx J x x J x
dx dxY x
I x I x
d dx I x x I x
dx dx
,
where the 1st row of the matrix above is for 0 and the bottom row for 0.
Remark 5: (i) One consequence of Theorem 1, due to the monodromy matrix 0B and
the initialization 0T P in the recurrence relation (1.4), is that the fundamental matrix
solution (1.3) to (2.3)-(2.4) captures the Frobenius solutions to the scalar Bessel equation
(2.1) in the 1st row of (3.15) and their quasi-derivatives in the 2nd row. (ii) Evidently the
44 matrix recurrence relation (3.12) has enabled two Bessel functions and their quasi-
derivatives to emerge by a novel means: namely, as products of 44 matrices. We now
demonstrate that our vectorization procedure applies equally well in implementing
Theorem 2.
16
Section 5: An Example of the Vectorization Procedure; Theorem 2
In this section we take note of the fact that the vectorization procedure, used to solve the
matrix recurrence relation (1.4) in Theorem 1, applies equally well to solve the
recurrence relation (1.6) in Theorem 2. As it turns out, Theorem 2 yields in the 1st row of
its fundamental matrix , ,X x linear combinations of the two Frobenius solutions
1 ,y x and 2 , ,y x obtained earlier by application of Theorem 1.
We find 2 1
2 2,n
xS 1,2,3,4,5,...,n while for 2,n the recurrence relation (1.6) yields
(5.1)
2
2 2 2
2 2 2
2 2
2 2 2
2
2 2 2
2 0
2 0,
2
2 0
a b c
a b d
a c d
b c d
where 2 2
2
2 2
.a b
Sc d
Similar to what we obtained in equation (3.4), we find that by
employing a vectorization procedure gives
(5.2)
01
2
2 2 2
2
2
2 1
2 2 0 2 2 4 2 22
2 2 22
2 10 0 0 0 1
1 2 1 1 0 0 0 0 01: ( ) .
0 0 0 04( 1) 2
0 0 0 11 2
S
M
a
bS M S
c
d
17
Interestingly, we find by our vectorization procedure employed in conjuction with
Theorem 2, the matrix key to this means of representation above is the same as before
when we applied our vectorization procedure in conjuction with Theorem 1 so
5
5
2
5
5
1 1
2 ( 1) 1
1 1 1
2 ( 1) 1
,
2 2
2 ( 1) 1
1 1 1
2 ( 1) 1
v v
v v
S
v v
v v
v v v
(5.3) 1 1 1 1 1
2 2 2 2 2 2 2 2 4 2 0( )( )( ) ( )j j j j j jS M S M M M M S
1
1
2 2 0
0
j
j N
N
M S
18
2 1
2 1
2 1
2 1
( 1) 1 1
2 ! ( 1) 1
( 1) 1 1 1
2 ! ( 1) 1
( 1) 2 2
2 ! ( 1) 1
( 1) 1 1 1
2 ! ( 1) 1
j j
j
j j
j j
j
j j
j j
j
j j
j j
j
j j
j v v
j v v
j v j v
j v v
j v v v
,
1,2,3,...,j
(5.4)
2 2 22
2 2
1
2 2 24 2 2
2 2
2 2 22
21
2 11 1
1,
4( ) 2
21
j
j
j j
j
j jM
j j
j j
j
j j
where again we have employed Mathematica to compute the matrix inverse.
Accordingly, the fundamental solution matrix (1.5) is readily calculated as
(5.5)
03 4 2
2
03 4
, , 2 2( , ) ( )
, , ( )
2 2
A j
j
j
x x x x
y x y xX x S x x S x
xy x xy x x x x x
19
2 2 2 2
2 2 2 2
0 0 0 0
2 2 2 2
2 2 2 2
0 0 0 0
1 1 1 1
2 2 2 2.
1 1 1 12 2 2 2
2 2 2 2
j v j j v j
j j j j
j j j j
j v j v j v j
j j j j
j j j j
x x x x
j x j x j x j x
Here 2 j and
2 j are given by
(5.6)
2 22 2
1 1, ,
2 ! ( 1) 2 ! (1 )
j jj j
j jj j
j jj j
and we have used
(5.7)
0
0
1
20
0 1 01 2 2exp ln ln .
0 0! ( )
2 2
n
A n
A
x x x x
x x P P xn x x x x
Comparison between (5.5)-(5.6) and (3.15) now reveals
(5.8)
1 2 1 2
3 4
3 41 2 1 2
1 1 1 1( , ) ( , ) ( , ) ( , )
, , 2 2 2 2( , ) ,
1 1 1 1, ,( , ) ( , ) ( , ) ( , )
2 2 2 2
y x y x y x y xy x y x
X xxy x xy x
x y x y x x y x y x
where the 1st row of ( , )X x may be represented in terms of Bessel functions as
20
(5.9)
2 2
3
2 2
1 12 ( 1) ( ) 2 (1 ) ( ) , 0
2 2( , ) ,
1 12 ( 1) ( ) 2 (1 ) ( ) , 0
2 2
J x J x
y x
I x I x
(5.10)
2 2
4
2 2
1 12 ( 1) ( ) 2 (1 ) ( ) , 0
2 2( , ) .
1 12 ( 1) ( ) 2 (1 ) ( ) , 0
2 2
J x J x
y x
I x I x
Remark 6: One consequence of Theorem 2, due to the monodromy matrix 0A and the
initialization 0S I in recurrence relation (1.6), is that the fundamental matrix solution
(1.5) to (2.3)-(2.4) captures linear combinations of Bessel Functions in the 1st row of (5.5)
with their corresponding quasi-derivatives in the 2nd row. Lastly we’ll sketch the
vectorization procedure as it may be applied to a general Frobenius-type problem of the
non-logarithmic cases.
21
Section 6: A Vectorization Procedure for the General Problem
For the general problem with ( )x being a 22 matrix,
(6.1) 2
0 1 2
1 1( ) ,X x X A A x A x X
x x
where 0
( ) n
n
n
x A x
is convergent for x a for some 0.a Recurrence (1.4) is
(6.2) 1
0 0 0 1 1 1 1
0
( ) ,n
n n n m m n n n
m
T B nI A T A T A T A T AT
1,n
where as in Theorem 1,
(6.3) 11
0 0
2
0,
0
rB P A P
r
and 0 .T P
If 1r and 2r are real with 1 2 ,r r then the assumption that (6.1) has a good spectrum is,
(6.4) 1 2 ,r r n 1,2,3,...,n
which rules out the logarithmic cases of Frobenius theory as required and hence the
vectorization procedure introduced in section 2 is applicable to solve (6.2). This fact
follows immediately from the lemma of Gantmacher and Balser in (2.10) as applied to
(6.2) since for 1,2,3,...,n the eigenvalues 1r n and 2r n of 0B nI differ from the
eigenvalues 1r and 2r of 0A because of assumption (6.4). Thus a unique solution of (6.2)
for the 22 matrix nT must exist and since the right hand side of (2.10) is immaterial for
the application of this lemma, successive solution for unique 1 2, , , nT T T is possible.
22
To illustrate the vectorization procedure in this general case, consider the first step for
1,n
(6.5) 1 0 0 1 1 0.T B I A T AT
Letting 0 0
0
0 0
,a b
Tc d
1 1
1
1 1
,a b
Tc d
and ,n n
n
n n
D EA
F G
0,1,2,3,...,n
then the matrix equation (6.5) corresponds to the four scalar equations
(6.6)
1 1 0 1 0 1 1 0 1 0
1 2 0 1 0 1 1 0 1 0
1 1 0 1 0 1 1 0 1 0
1 2 0 1 0 1 1 0 1 0
( 1)
( 1).
( 1)
( 1)
a r D a E c D a E c
b r D b E d D b E d
c r F a G c F a G c
d r F b G d Fb G d
Writing (6.6) in the form of a 44 matrix equation we see that the analog of eq. (3.4) is
(6.7) 1 1 1 0 ,M T T
where again
0
0
0
0
0
,
a
bT
c
d
1
1
1
1
1
,
a
bT
c
d
and in this general case we find
(6.8)
1 0 0
2 0 0
1
0 1 0
0 2 0
1 0 0
0 1 0
0 1 0
0 0 1
r D E
r D EM
F r G
F r G
,
1 1
1 1
1
1 1
1 1
0 0
0 0:
0 0
0 0
D E
D E
F G
F G
.
Hence, it follows that
(6.9) 1
1 1 1 0.T M T
23
The fact that 1M is invertible follows (under the assumption of good spectrum) directly
from the existence of a unique solution for 1T of equation (6.5). Likewise, in the thn step,
recurrence relation (6.2) converts by our vectorization procedure to
(6.10) 1
0
, 1,2,3,...,n
n n n m m
m
M T T n
where
(6.11) ,
n
n
n
n
n
a
bT
c
d
1 0 0
2 0 0
0 1 0
0 2 0
0 0
0 0,
0 0
0 0
n
r n D E
r n D EM
F r n G
F r n G
(6.12)
0 0
0 0:
0 0
0 0
n m n m
n m n m
n m
n m n m
n m n m
D E
D E
F G
F G
A closed form representation for nT is then
(6.13) 1 1
1 1
0 0
, 1,2,3,...,n n
n n n m m n n m m
m m
T M T M T n
which formulates the 41 vector nT explicitly as a sum of products of 44 matrices with
41 vectors whose representations are all readily obtained from this section. Thus, under
the assumption of good spectrum of system (6.1), our vectorization procedure indeed
applies in the general case. Next, we’ll give a brief history of the Bessel equation and the
power series method.
24
Section 7: Some History on the Bessel Equation and the Power Series Method
The Bessel equation 2 2 2 0x u xu x a u takes its name from the German
astronomer F. W. Bessel who in 1824, while studying the perturbations of the planets,
began the development of the now extensive theory of its solutions (See [19].). In
Bessel’s treatment, he assumed the parameter a to be an integer and managed to obtain
integral definitions of the Bessel functions. Schlömilch and Lipschitz later named these
functions after Bessel [56, p. 319]. Interestingly, the now standard power series
representations of the Bessel functions, as given in (4.3)-(4.6), do not appear in Bessel’s
1824 seminal paper and they are also missing from the original papers of Frobenius and
Fuchs (See [38], [41].). Given that the power series solution ansatz was originated by L.
Euler [3, p. 204] and thus it predates the work of Bessel, Frobenius, and Fuchs, these
omissions are notable. In fact, the power series formulas for Bessel functions arose first
not from the Euler/Frobenius/Fuchsian theory but from the integral representations and
thus it’s these integral representations that appear in the early books (compare E.C.J. von
Lommel [73], C.G. Neumann [81], and G.N. Watson [101, Chapter 2]). Some special
instances of series expansions of Bessel functions were however generated even before
Bessel’s 1824 paper through investigations to such varied topics as the oscillations of
heavy chains and circular membranes, elliptic motion, and heat conduction in cylinders
(In [18], D. Bernoulli, in 1738, obtained an open form series expansion of a Bessel
function of order 0 or [35] where L. Euler, in 1764, generated a Bessel series while
investigating vibrations of a stretched circular membrane.). See the table below.
25
I - Solutions of a scalar Bessel equation are obtained as a power series near 0x
using the power series method.
II - A 22 matrix power series method for 2nd order equations is given.
III - A 22 matrix form of the Bessel equation is given.
Table 1 - The Bessel Equation, Power Series Method, and Matrix Formulations
Author; Year; Reference I II III
F.W. Bessel; 1824; [19] No No No
L. Fuchs 1866; 1868; [40]-[41] No No No
C.G. Neumann; 1867; [81] No No No
E.C.J. von Lommel; 1868; [73] No No No
G. Frobenius; 1873; [38] No No No
A. Gray and G.B. Matthews; 1895; [52] Yes, p. 7-12 No No
L. Schlesinger; 1897; [91] No No No
L. Schlesinger 1900; 1904; 1922; [89] Yes, p. 274-279 Yes, p. 132-142 No
A.R. Forsythe; 1902; 1959; [37] Yes, p. 100-102 No No
H.F Baker; 1902; [8] No Yes, p. 335 Yes, p. 348
Whittaker and Watson; 1902; 1915; 1920;
1944; [104] Yes, p. 197-202 No No
A.R. Forsythe; 1903 3rd Ed.; [36]
Yes, p. 176-186 in
the 6th Ed. No No
N. Nielsen; 1904; [82] Yes, p. 4-5 No No
G.N. Watson; 1922; 1944; [101] Yes, p. 38-42; 57-63 No No
R. Courant & D. Hilbert; 1924; 1931; [27] Yes, p. 260; 418 No No
E.L. Ince; 1926; [66] Yes, p. 403-404 Yes, p. 408-415 Yes, p. 415
A. Erdelyi; 1953; [32] Yes, p. 4-5 No No
Lappo-Danilevsky; 1936; 1953; [72] No Yes, p. 143 No
Coddington and Levison; 1955; [25] No Yes, ch. 4, p. 119 No
Abramowitz and Stegun; 1964; [1] Yes, p. 360; 437 No No
E. Hille; 1969; [56] Yes, p. 319-320
Yes, p. 188-191;
239-250 No
W. Balser; 2000; [9] Yes, p. 23 Yes, ch. 2, p. 19 No
Olver et al Nist; 2010; [84] Yes, p. 217 No No
26
So it appears that the power series method originated by L. Euler [3] was not applied to
the Bessel Equation until some years after the original papers of Frobenius and Fuchs,
(See [38], [40]-[41].). As indicated in the table, the earliest application of the
Euler/Frobenius/Fuchsian power series method to the Bessel Equation that I could find
was in the 1895 book of Gray and Matthews [52]. Also, as revealed by the above table,
only two authors in this literature search appear to have written the Bessel equation in a
matrix form, first Baker in [8] and then later Ince in [66]. Note though that neither of
these authors attempted to apply their 22 matrix forms of the power series method to
the matrix form of the Bessel equation so as to obtain a 22 fundamental matrix as given
in this paper in (3.15). This fact appears to stem from the computational difficulty of
generating the solution matrix X of (2.10) for use in implementing the matrix recurrence
relations (1.4) or (1.6), a matter not addressed by Schlesinger [89], Baker [8], Ince [66],
Lappo-Danilevsky [72], Coddington and Levison [25, ch. 4], Hille [56], or Balser [9, ch.
2], all who formulated a 22 matrix power series formulation of the Euler/Frobenius/
Fuchsian theory. At least it appears that none of these authors provide a nontrivial
example of a matrix equation for which the recurrence relations (1.4) or (1.6), in
Theorems 1 and 2, were successfully solved to yield an explicit representation of the
fundamental matrix solution in the form of (1.3) or (1.5). A search of the literature leads
this author to believe that the development of a general method to properly resolve the
computational difficulty in (1.4) and (1.6), associated with the 22 matrix power series
method, has indeed been left undone. This is further supported by a remark of Ince
27
[66, p. 415], where in specific reference to the matrix form of the Bessel Equation (9)-
(10), with 1, he states, “The scope of the matrix method is very wide, but its
successful application demands a knowledge of theorems in the calculus of matrices
which cannot be given here.” Likewise, after Hille furnishes his solution ansatz to the
matrix equation ( ) ( ) ( )w z F z w z involving a regular singular point at the origin, and
defines his “resolvent of the commutator”, he states, “Thus the formal solution is well
defined, if not explicitly known”, [56, p. 233]. It should be noted though that two authors
did however come close to obtaining an explicit solution of the 22 matrix recurrence
relation. Both Lappo-Danilevsky in [72] and then later Hille in [56] proposed highly
theoretical solutions to the 22 matrix recurrence relation using a 22 matrix theory.
Lappo-Danilevsky’s theory appears to be invoked in Hille’s resolvent of the commutator
[56, p. 233], which, in principle, is a matrix inverse operator that if one could explicitly
obtain it, then it would theoretically yield the general solution to the 22 matrix
recurrence relation in (1.4) or (1.6) or (6.2); and, in particular, the solution (3.15)
obtained here would be expressible as a product of suitable 22 matrices. The fact
though that there is no implementation to any specific examples of the Lappo-Danilevsky
theory or the Hille theory in their works or in any of the works by the later authors who
also gave a 22 matrix formulation of the Frobenius theory does indicate that an explicit
solution to the 22 matrix recurrence relation that could be implemented in the general
case as has been given here in Section 6 has heretofore eluded previous authors.
28
Summary of Results
In the literature, prior matrix treatments of Frobenius theory, like Theorem 1 and 2, make
use of coupled matrix recurrence relations that pose considerable difficulties in their
implementation. The new idea outlined in this paper, our vectorization procedure,
appears to provide a rather elegant means to reformulate the matrix recurrence relations
of both Theorem 1 and 2 into equivalent recurrence relations that require only
computation of well-defined matrix products and sums. It can also be noted that our
vectorization procedure seems to generalize and apply to the 3rd order, 4th order, … , thn
order Frobenius type problems of the non-logarithmic cases and near regular singular
points by generating equivalent recurrence relations that require only matrix
multiplications and sums involving 99 matrices, 1616 matrices, …, 2n 2n matrices
respectively.
29
Chapter 2: Sturm-Liouville Equations with Irregular Singular Points
The 41 x Potential; Irregular Singular Points at 0x and x
Chapter 2 Introduction: This chapter contains both original methods and results for
Sturm-Liouville equations with irregular singular endpoints. The new methods presented
here enable the determination of principal and canonical non-principal solutions near an
LP irregular singular endpoint. A feature of this new approach is the factoring away of
strongly singular controlling factors of solutions so that by the imposition of initial
conditions and terminal conditions at the LP irregular singular endpoints, iteration
towards Volterra integral equations of the second kind yield the remaining dependent
factors of solutions. The methods are carried out for a 2nd order Sturm-Liouville equation
with strongly singular 41 x potential and with LP irregular singular points at 0x and
.x Formal power series representations of solutions are thereby obtained, generated
about the irregular singular points, 0x and ,x and these solutions have
asymptotics that conform to those generated by application of the prior methods of
Poincaré [86], Thomé [94], Birkhoff [20], Erdelyi [30], and Jörgens [67]. Existence and
uniqueness results are established and asymptotics near 0x and x are given for
two linearly independent solutions generated about each singular endpoint. Several
connection problems are addressed. Considerations of the more general 1 Bx potential
are briefly discussed.
30
Section 8: The 41 x Potential SL Equation and Irregular Singular Point 0x
In this section, joint work [43] by C. Fulton and J. Mandelkern that is to be submitted for
publication is given. In this aforementioned work, the authors give a fundamental system
of solutions obtained near the irregular singular point 0x for the Sturm-Liouville
equation
(8.1) 4
1y y y
x , 0 x ,
as well as establish certain properties of these solutions. First to establish that 0x is
indeed an irregular singular point, we remind the reader that ( ) ( ) 0y P x y Q x y has
a regular singular point at 0x x when either of ( )P x or ( )Q x is singular at 0x x and
where
(8.2) 201 2 0 3 0
0
( ) ( ) ( )( )
PP x P P x x P x x
x x
,
and
(8.3) 0 12 3 02
0 0
( ) ( )( ) ( )
Q QQ x Q Q x x
x x x x
,
with not all 0P , 0Q , and 1Q equal to zero [44]. Here we see that with equation (8.1), on
account of the strongly singular potential 4
1( ) ,q x
x 0x is a singular point that does
not conform to this definition and hence it is an irregular singular point. Moreover,
regarding similar such integer power potentials, the 2nd order Sturm-Liouville equation
31
(8.4) ( )y q x y y , 0 x ,
has an irregular singular point at 0x whenever ( ) 1 ,Nq x x with 3,4,5,...N .
When the SL equation (8.4) has a singularity at 0x of type LP/N (Limit Point/Non-
Oscillatory) (See [103].), it can be expected that there exists a principal solution near
0,x ( , ),py x which satisfies the properties in [49] or [51]:
(i) 2 0( , ) (0, )py x L x for 0 (0, )x and .
(ii) ( , )py x is entire in for each fixed (0, ).x
(iii) ( , ) ( , )p py x y x (0, )x and .
Additionally, it can be expected that there will exist a particular choice of non-principal
solution, ( , ),npy x which satisfies properties (ii)-(iii) only. In this section, we obtain
( , )py x and ( , ),npy x for an SL equation with a strongly singular potential and an LP/N
irregular singular point at 0.x The crux of our method’s implementation near 0x
involves factoring away independent singular behaviors of these solutions by the use
of suitable changes of variables. The remaining non-singular dependent factors of
( , )py x and ( , )npy x are then obtained by the solution of Volterra integral equations of
the 2nd kind. Unlike the formal power series solution methods of Poincaré, Thomé,
Burkhardt, Jörgens, and Erdelyi, which furnish asymptotics of true solutions, our method
yields, by recursion, the true solutions for which these asymptotics correspond to.
The solution of SL equations about singular endpoints was initiated by G. Frobenius in
32
[38] and since then much work in this area has been done. For example, in the case that
0x is a regular singular point (RSP) in addition to being LP/N, the solution ( , )py x
can be taken to be the first Frobenius solution near 0x (the solution with larger indicial
root), a result established in Fulton [48] and Fulton and Langer [42], together with
examples involving Bessel’s equation, the H-atom, and the associated Legendre’s
equation. The normalization of the solution with respect to is achieved in these cases
without initial conditions at 0,x by fixing suitable normalizations of the Frobenius
solutions. The question as to how such normalizations for ( , )py x can be chosen for the
cases when 0x is of LP/N or LC/N (Limit Circle/Non-Oscillatory) type but not a RSP,
remains for the most part completely open [47, p. 39]. For the LC/N case ( ) ,c
q xx
1 2 , it was shown by Atkinson and Fulton in [7], that when ( , )py x satisfies the
above properties (i)-(iii), it could be normalized by the requirement ( , )py x x higher
powers of ,x where the higher powers are not necessarily integral powers. We now wish
to develop the approach of this section for an SL equation with an LP/N irregular singular
endpoint at 0x and a strongly singular potential.
We now consider the SL equation of form (8.4) with LP/N irregular singular point 0,x
(8.5) 4
1, 0 ,y y y x
x
and show that independent singular factors of two uniquely defined solutions,
( , )py x and ( , ),npy x may be factored away leaving ODEs for the remaining factors of
33
these solutions. To ensure that ( , )py x still satisfies (i)-(iii), specific ICs for the
remaining dependent factors must be imposed at the singular endpoint 0.x This
approach appears to be new; namely, the imposition of ICs at an irregular singular
endpoint, as is done in this dissertation, so as to define a principal solution. We now
demonstrate the method. First we identify the singular behavior of ( , )py x and
( , ).npy x For the case 0 in equation (8.1), the changes of variables
(8.6) 1
2 ( )y x Y x and 1
( )Y t Yx
reduces
(8.7) 4
10y y
x
to the modified Bessel equation of order 1
,2
(8.8) 2 21( ) ( ) ( ) 0.
4t Y t tY t t Y t
Since
(8.9) 1 2
1 1( ) exp( ), ( ) exp( )Y t t Y t t
t t
is a fundamental set of solutions to (8.8), it follows that
(8.10) 1
1( ) xy x xe and 1
2 ( ) xy x xe
are two linearly independent solutions of (8.7).
34
As with Frobenius solutions of SL equations about RSPs, it can be expected that the
asymptotic form of a solution of equation (8.5) will be independent of . Thus (8.10)
suggests that the two changes of variables
(8.11) 1
( , ) ( , )xy y x xe w x
and 1
( , ) ( , )xy y x xe v x
can be applied to (8.5) to generate ODEs for the remaining dependent factors of
( , )py x and ( , ).npy x Note that a second independent motivation for the use of the
substitutions in (8.11) comes from an application of the method of variation of
parameters to the “homogeneous” and “nonhomogeneous” equations
(8.12) 4
10,y y
x
and
(8.13) 4
1.y y y
x
This procedure yields the Voltera integral equation
(8.14)
0
1 1 1 1 1 1
( , ) ( , ) ,2
x
x x x t x t
x
xty x E xe F xe e e y t dt
after which the authors then used this form of solution to surmise that the substitutions in
(8.11) could be utilized so as to factor away the singular factors 1
xxe and 1
xxe
appearing in the first two terms above in (8.14). We find from the changes of variables
(8.11) applied to (8.5) that
35
(8.15) 2
2 2( , ) ( , ) ( , ) 0,w x w x w x
x x
and
(8.16) 2
2 2( , ) ( , ) ( , ) 0.v x v x v x
x x
It may now be expected that certain initial conditions at 0x must be imposed on
( , )w x and ( , )v x so that ( , )py x satisfies (i)-(iii) and ( , )npy x satisfies (ii)-(iii) only
as required. To discover such required ICs at 0,x we convert ODEs (8.15) and (8.16)
into equivalent Volterra integral equations of the 2nd kind. To obtain the Volterra integral
equations equivalent to (8.15) and (8.16), we apply variation of parameters arguments on
the “homogeneous” and “nonhomogeneous” equations
(8.17) 2
2 2( , ) ( , ) 0w x w x
x x
homogeneous ( , )w x equation
and
(8.18) 2
2 2( , ) ( , ) ( , )w x w x w x
x x
nonhomogeneous ( , )w x equation
as well as
(8.19) 2
2 2( , ) ( , ) 0v x v x
x x
homogeneous ( , )v x equation
and
(8.20) 2
2 2( , ) ( , ) ( , )v x v x v x
x x
nonhomogeneous ( , )v x equation
The result of which is stated below as Lemma 1.
36
Lemma 1: If ( , )w x and ( , )v x are twice differentiable with respect to ,x 0 0,x then
I. Every solution to (8.15) is also a solution to the Volterra integral equation
(8.21)
0
2 222
( , ) 1 ( , ) ,2
x
x x t
x
tw x A Be e w t dt
for some choice of the constants A and B and vice versa.
II. Every solution to (8.16) is also a solution to the Volterra integral equation
(8.22)
0
2 222
( , ) 1 ( , ) ,2
x
x x t
x
tv x C De e v t dt
for some choice of the constants C and D and vice versa.
PROOF: Since
2
1, xe
is fundamental set of solutions to (8.17) having Wronskian
determinant, 2
22 ,xW x e variation of parameters, as applied to (8.18), yields
(8.23)
2 2
1 2( , ) 1 1 .x x
particular solution
w x A B e u u e
Here we point out that for 1
( , ) ( , )xpy x xe w x
to satisfy square-integrability
property (i) near 0x , we must take 0.B It still remains to be shown that
(8.24)
0
2 2 22
1 21 1 ( , ) .2
x
x x t
xparticular solution
tu u e e w t dt
To this end, 1u and 2u are computed using the relations
37
(8.25)
11 ,
Wu
W 2
2 ,W
uW
where
2
1 2
2
0,
2
x
x
eW
w x e
2
1 0,
0W
w
and
222 .xW x e
Integration from 0x to x of 1u and 2u as specified in (8.25) now yields
(8.26)
0 0
2
11 ( ) ( , ) ,
2
x x
x x
W tu t dt w t dt
W
and
(8.27)
0 0
222
2 ( ) ( , ) .
x x
t
x x
Wu t dt t e w t dt
W
Lastly we insert (8.26) and (8.27) into (8.24) and (8.21) has been established. Now let
( , )w x be a twice differentiable with respect to x solution to (8.21). Computation of
( , )w x and ( , )w x shows that ( , )w x satisfies (8.15) so I. is now established. The
equivalence of ODE (8.20) to the Volterra integral equation (8.22) is obtained in the same
manner. The question arises as to whether we can replace 0x in Lemma 1 by 0; and
more particularly, whether we can take 0B and prove that there exists solutions of
(8.21) and (8.22) having continuity of ( , ),w x ( , ),w x ( , ),v x and ( , )v x as 0 .x
To this end we first establish a lemma.
38
Lemma 2: I. If a solution ( , )pw x of (8.15) that is bounded in [0, ],b then for some
0,A ( , )pw x will satisfy the Volterra integral equation
(8.28)
2 22
0
( , ) 1 ( , ) ,2
x
x tp p
tw x A e w t dt
and the initial conditions
(8.29) 0
lim ( , )px
w x A
and 0
lim ( , ) 0px
w x
.
PROOF: Putting 0B and 0 0x into (8.21) gives
(8.30)
2 22
0
( , ) 1 ( , ) .2
x
x tt
w x A e w t dt
Since 0 ,t x
2 22 2
12 2
x tt b
e
(0, ],x b the integral in (8.30) is necessarily finite
under the assumption that ( , )w x M (0, ]x b and . To verify that the
initial conditions (8.29) hold, observe that
(8.31)
2 2 2 22 2 2
0 0 00 0 0
22 22
4
0
2 220 0 0
lim ( , ) lim ( , ) ( , ) lim ( , )2 2 2
( , ) ( , )2 2lim lim lim ( , ) .42
x x x
x t x t
x x x
x
tx
L
x x xx x
t t tw x A e w t dt w t dt A e w t dt
t xe w t dt e w xx
A A A w x Ae e x
Observe also that
39
(8.32)
22
2 22
0
2 2/0 0 0
0
( , )
lim ( , ) lim 1 ( , ) lim 0,2
x
tx L
x tx
x x x
t e w t dtd t
w x e w t dtdx x e
where we have used the assumption that ( , )w x is bounded on 0, .b Here we have
also used the Liebniz rule below and L’Hôpital’s rule. The Lemma 2 is now proven.
(8.33)
( ) ( )
( ) ( )
( , ) ( , ) ( , ( )) ( ) ( , ( )) ( )
x x
x
x x
df x y dy f x y dy f x x x f x x x
dx
Liebniz rule
We will now show the existence of principal and non-principal solutions to (8.5) near
0.x Due to the singular controlling factors identified in (8.11), solutions of the original
equation (8.5) cannot be uniquely determined by specification of initial conditions at
0.x On the other hand, we obtained two different yet related ordinary differential
equations, (8.15) and (8.16), whose solutions have a benefit in that they do not inherit the
strongly singular controlling factors that must necessarily appear in any solutions to (8.5).
For this reason, we now turn our attention to the relatively simpler equation (8.15), which
is restated below as equation (8.34) for the convenience of the reader, and prove that it
has a unique solution provided that suitable initial conditions are imposed. Namely, the
unique solution to
(8.34) 2
2 20,w w w
x x
40
will be fixed by imposing the initial conditions for all ,
(8.35) 0
lim ( , ) 1,x
w x
0
lim ( , ) 0.x
w x
We now address the existence of a solution to IVP (8.34-8.35). Using variation of
parameters on the homogeneous and nonhomogeneous equations
(8.36)
and the fundamental system of the homogeneous equation, the initial value
problem (8.34)-(8.35) converts to the equivalent Volterra integral equation
(8.37)
We have the following Theorem on existence of a solution of (8.37). The solution is
unique and a proof of the uniqueness will also be given in this section.
Theorem 3: There exists a bounded solution in 0[0, ]x of the Volterra integral equation
(8.37) for every 0 0,x which satisfies ICs (8.29) with 1.A Accordingly, this solution
is also a solution of the differential equation (8.34). Moreover, this solution is
continuous in x and in and entire in for each fixed x .
PROOF: Anticipating that the solution satisfying (8.34)-(8.35) is entire in , we
make use of an ansatz with roots from the study of celestial mechanics [56, p. 96].
(8.38)
2
2 20,w w
x x
2
2 2,w w w
x x
2/ ,1xe
2 22
0 0
( , ) 1 ( , ) ( , ) 1 1 ( , ) .2
x x
x tt
w x K x t w t dt e w t dt
[0, ) (0, )
( , )w x
0
( , ) ( ) .n
n
n
w x a x
41
Inserting (8.38) into the differential equation (8.34) gives
(8.39)
Also, inserting (8.38) into the two initial conditions (8.35) gives
(8.40) 0 0( ) 1, ( ) 0a x a x for all
and for 1,n
(8.41) (0) 0, (0) 0.n na a
Equating like powers of to zero in (8.39) and employing the initial conditions in (8.40)
and (8.41) thus gives for 1,n the coupled sequence of initial value problems,
(8.42) (0) 0, (0) 0.n na a
For , the initial value problem (8.42) converts using variation of parameters
once more to the integral equation
(8.43)
To prove convergence of the series in the original ansatz (8.38), we observe that
(8.44) for
and therefore
(8.45) for
Using (8.43) and (8.45), an induction argument gives for 1,2,3,...n
120 1
2 2( ) ( ) ( ) 0.n n
n n n
n n
a x a x a xx x
x (0, ),
12
2 2( ) ( ) ( ),n n na x a x a x
x x
1,2,3,...n
2 22
1 1
0 0
( ) ( , ) ( ) 1 ( ) .2
x x
x tn n n
ta x K x t a t dt e a t dt
2 2 2 2
0 1t x
x t xte e
0 ,t x
2 2
1 1x te
(0, ].t x
42
(8.46)
For the induction, observe that for we have from (8.43) and (8.45)
(8.47)
Now assuming (8.44) for ,n observe that (8.43) yields
(8.48)
Accordingly, using the bound (8.46) in (8.47) we find
(8.49)
Hence for we have the majorant for
(8.50)
Here R is a fixed positive real number. Since the series on the RHS of (8.58) is
convergent by the ratio test, it follows by the Weierstrass M-test [4, p. 413] that the series
defining ( , )w x is uniformly convergent in ( , )x for all
Hence ( , )w x is continuous in ( , )x in this domain. Moreover, for fixed the
uniformity of convergence for R implies by the Weierstrass Theorem [4, p. 174]
that 0( , )w x is analytic for .R Since b and R are arbitrary, we have that the series
defining ( , )w x is continuous in ( , )x for all and all and entire in
3
( ) .6 !
n
n n
xa x
n
1,n
2 3
1
0
( ) .2 3!
xt x
a x dt
2 3 3( 1)
1 1
0
( ) .2 6 ! 6 ( 1)!
x n n
n n n
t t ta x dt
n n
3
0 1
( , ) ( ) 1 .!6
n
n
n nn n
xw x a x
n
( , ) [0, ] : ,x b R ( , ),w x
3
1
( , ) 1 .!6
n
nn
Rbw x
n
( , ) 0, : .x b R
0 [0, ],x b
[0, ),x ,
43
for each fixed Next we need to establish that ( , )w x is a solution of the
Volterra integral equation (8.37). Since the series in (8.38) is uniformly convergent on
compact subsets of we have by the continuity of in (0, ]t x that for
fixed
(8.51)
uniformly for (0, ].t x Hence it follows from [5, p. 399] that
(8.52)
On the other hand, interchanging the order of summation and integration gives
(8.53)
as .N Since 0 1a as determined in (8.39), it follows from (8.52) and (8.53) that
(8.54)
Moreover, this holds for all and all Since ( , )w t is bounded by (8.50)
uniformly for [0, ],t b Lemma 2 applies; hence with the solution ( , )w x in
(8.38) satisfies the initial conditions
(8.55) and
So Theorem 3 establishes the existence of a solution to the initial value problem (8.34)-
0, .x
[0, ) , ( , )K x t
x 0,
0
lim ( , ) ( ) ( , ) ( , )N
n
nN
n
K x t a t K x t w t
0 00 0
lim ( , ) ( ) ( , ) ( ) .
x xNn n
n nN
n n
K x t a t dt K x t a t dt
1 1
1 0
0 0 00 0
( , ) ( ) ( , ) ( ) ( ) ( , )
x xN N Nn n n
n n n
n n n
K x t a t dt K x t a t dt a x w x a
0
( , ) 1 ( , ) ( , ) .
x
w x K x t w t dt
(0, )x .
1A
(0, ) 1w (0, ) 0.w
44
(8.35), or equivalently, the existence of a solution to the Volterra integral equation (8.37).
Next we establish that this solution is unique for all ; of course, it is well-known
that the principal solution is unique only up to a constant multiple, and by selecting
0 1a in the ansatz (8.38), or 1A in Lemma 2, equation (8.28), the constant factor is
fixed. First we may observe that the standard Picard theorem as well as the standard
Carathéodory existence and uniqueness theorem [25, p. 3, 43] does not apply to the IVP
(8.34)-(8.35). Writing equation (8.34) in the system form
(8.56)
where
(8.57)
the standard choice of a Lipshitz constant for which
(8.58)
in some domain of
(8.59)
is as given in [65, p. 3-4] or [2, p. 134]:
(0, )x
2
1
22 2 12
0 1
( ) ( , , )2 2 2 2
wf
W A x W W f x Wf w w
x x x x
1
2
( , )( , ),
( , ) ( , )
w xw xW
w x w x
0
1 2 1 20
sup ( ) ,x x
A x W W L W W
,f
0: 0, : ,R x W W B
45
(8.60) .
But for the IVP (8.34)-(8.35), this is not bounded as 0 ,x and hence fails to define a
Lipshitz constant on Similarly, in the case of the Carathéodory initial value
problem at 0x [25, p.43], uniqueness of the solution of the IVP (8.34)-(8.35) would
require that
(8.61)
for some Lebesque-integrable function ( ),m x that is
(8.62)
But again, this is not the case since the function is not integrable near 0.x
Hence the Carathéodory Existence Theorem too cannot be applied to establish
uniqueness of the bounded solution near 0.x To prove the uniqueness of the solution
(8.38) obtained in the proof of Theorem 3, we instead follow an argument similar to that
used in Fulton, Pearson, Pruess [46, p. 195, Thm. 1], by reliance on a Banach fixed point
theorem. Namely, we prove the following theorem:
Theorem 4: For sufficiently small the Volterra integral equation (8.37) has a
unique solution in satisfying the initial conditions
0 0
2,0 0, 2
2 2: 2 2 sup max 2 sup max 0,1, , ( 1)i
i jx x x xji jW B W B
fL M
w x x
00, .x x
0 0
2 1 220 0
2 2sup ( , ) sup max , ( ),
x x x x
W B W B
f x W w w w m xx x
0
0
( ) .
x
m x dx
2
2 2
x x
0 0,x
00, x
46
(8.63) and
PROOF: Consider the Banach space
(8.64) := 0[0, ] : ( )C x f f x is continuous on 0[0, ] ,x
the space of complex-valued functions which are continuous on the closed interval
0[0, ],x with norm
(8.65) 00
sup ( ) .x x
f f x
The metric associated with this norm on is
(8.66) ( , ) : .d f g f g
Now consider the subset of defined by
(8.67) : : (0) 1 .f f
To see that is a closed subset of , let nf and f such that ( , ) 0.nd f f
Since (0) 1nf for all ,n and
(8.68) 00
(0) (0) sup ( ) ( ) ( , ) 0,n n nx x
f f f x f x d f f
it follows that (0) 1.f Hence is a closed set in the Banach space and is therefore
itself a complete metric space with the same metric. Now consider the mapping
:T defined by
(8.69)
2 22
0 0
( )( ) 1 ( , ) ( ) 1 1 ( )2
x x
x tt
Tf x K x t f t dt e f t dt
To see that T maps into , we observe that since 00, ,f C x Lemma 2 gives
(0, ) 1w (0, ) 0.w
47
(8.70) 0
lim( )( ) 1.x
Tf x
To establish that (8.69) has a unique solution ,f we now prove the following claim.
Claim: For sufficiently small 0 0,x T is a contraction map on the metric space ( , ).d
To prove this claim, observe that
(8.71) 0
( )( ) ( , ) ( ) ( ) .
x
Tf Tg x K x t f t g t dt
Hence
(8.72) 3
0
0
( )( ) ( , ) .6
xx
Tf Tg x f g K x t dt f g
It follows using the definition of the metric (8.66) on and on , that
(8.73)
3
0( , ) ( , ).
6
xd Tf Tg d f g
Hence by choosing 0x sufficiently small so that
(8.74)
3
01,
6
x
it follows that T is a contraction mapping on . The Banach fixed point theorem
[71, p. 300] therefore applies to with the above metric (8.66), and so there is a unique
element f satisfying
(8.75) 0
( ) ( )( ) 1 ( , ) ( ) ,
x
f x Tf x K x t f t dt
48
and
(8.76) (0) 1.f
It remains to show that the fixed point, ( ),f x is differentiable and satisfies the additional
initial condition in (8.63),
(8.77) 0
(0) lim ( ) 0.x
f f x
Differentiation of (8.75) gives
(8.78)
22
22
0 0
( , )( ) ( ) ( ) ,
x x t
x
df K x t t ex f t dt f t dt
dx x x e
which is continuous for all 0(0, ].x x To extablish that ( )f x also satisfies the initial
condition (8.77) at 0,x observe from L’Hôpital’s rule
(8.79)
22
22
0
2 2 220 0 0
2
0
( )( )
lim ( ) lim lim2 2
( )lim 0,
2 2
x
t
xL
x x xx x x
x
t e f t dtx e f x
f xx e e xe
x f x
x
Since ( ) 1f x in this limit by (8.76). This completes the proof of Theorem 4.
Corollary 1: (i) The solution ( , )w x in (8.37) of Theorem 3 is equal to the fixed point
( )f x in Theorem 4, and also represents the unique solution of the IVP (8.34)-(8.35) on
[0, ).x (ii) ( , )w x is entire in .
49
PROOF: (i) For each fixed , ( ) ( , )f x f x in (8.74) and ( , )w x in (8.38)
(with ( )na x defined by the recursive scheme (8.43) are solutions of the IVP (8.34)-(8.35)
at least for 0[0, ].x x Hence ( , ) ( , )w x f x for all 0[0, ],x x and is unique by
virtue of Theorem 4. For the unique extension of the solution ( , )w x to 0[ , ),x x for
a fixed value of and a fixed 0[0, ],x x consider the IVP (8.80)-(8.81)
(8.80) 2
2 2( , ) ( , ) ( , ) 0,z x z x z x
x x
(8.81) ( , ) ( , ), ( , ) ( , ).z x w x z x w x
Standard arguments for IVPs now apply to establish uniqueness of the solution, ( , ),z x
of the IVP (8.80)-(8.81) for all (0, )x (See [93, p. 182, Thm. 9.1] or [29, Vol. II,
Thm. 3, p. 1281].). But since the solution ( , )w x constructed in (8.38) also satisfies the
IVP (8.80)-(8.81), it follows from the uniqueness for this IVP that ( , ) ( , )z x w x for
all (0, ).x (ii) For the second part, it was already established by application of the
Weierstass M-test in Theorem 3 that ( , )w x as defined by (8.38) and (8.43) is entire in
. This completes the proof of Corollary 1.
We now consider the v equation (8.16). This equation has 0x as an irregular
singular point of Poincaré rank 1; however, the essential singularity exp(1 )x in the
y solution has been factored out, so that the v equation will have solutions which are
bounded and continuous at 0.x Multiplying both sides of (8.16) by the integrating
50
factor integrating factor 2
2( ) ,xp x x e (8.16) becomes 2 2
2 2 0.x xx e v x e v
For
0 this equation has the general solution
(8.82) 1 2
1( ) 1 exp 2 ,
2v x Av Bv A B x
and for this choice of fundamental system we have
(8.83) 2
1 2exp 2 ( , ) 1.xx x W v v
In this case we have
(8.84) 2
0 01
( ) 1lim lim exp( 2 ) 0,
( ) 2x x
v xx
v x
so 2 ( )v x is the principal solution at 0x and 1( )v x is a non-principal solution. Under
the changes of variables in (8.11) it will be observed that solutions of the wequation
(8.15) and the v equation (8.16) are related by
(8.85) ( , ) exp( 2 ) ( , ).v x x w x
Accordingly, if ( , )w x is taken as the small (principal) solution in (8.11) and found in
Theorems 3 and 4, then the small (principal) solution of (8.16) may be defined as
(8.86) ( , ) : exp( 2 ) ( , ).pv x x w x
For fixed , there exists ( ) 0a a such that there no zeros of ( , )w x for
[0, ( )].x a We can therefore make the following definition of a “large” (non-
principal) solution near 0x for all (0, )x and all as follows:
51
Definition 2: For each fixed , let ( )a be chosen as above, and define, in terms of
the “small” solution, ( , ),w x a second linearly independent solution by the reduction of
order formula:
(8.87)
2( ) ( )2
( ) 2 2 22 2
1( , ) : 2 ( , ) 2 ( , )
( , )( , )
a a sx
a ps
x xp
ev x v x ds e w x ds
s w ss e v s
for [0, ( )].x a Here, for Im 0, ( , )w x and ( , )pv x are complex-valued
solutions of (8.15) and (8.16); but it is easy to verify that for a complex-valued solution of
(8.16), the reduction of order formula generates another complex-valued solution to
(8.16). The solution defined in (8.87) is a uniquely-defined solution of (8.16) by virtue of
the uniqueness of the solution ( , )w x from Theorem 4. For each fixed , we can
extend ( ) ( , )av x to (0, )x by choosing as initial conditions for any (0, ( )],x a
values of the above solution, ( ) ( , )av x and ( ) ( , )av x (See D.S. [29, p. 1281, Thm.
3].) We can then make a unique definition of a “large” solution of (8.16) for (0, )x
and all , by putting for each :
(8.88) ( , ) :v x uniquely-extended solution ( ) ( , )av x to (0, ).
We now prove the following theorem establishing properties of ( , ) :v x
52
Theorem 5: The solution defined by (8.87)-(8.88) for ( , ) (0, ) ,x satisfies the
following properties:
(i) 0
lim ( , ) 1x
v x
.
(ii) 0
lim ( , ) 0x
v x
.
(iii) ( )( ( ), ) ( ( ), ) 0av a v a .
(iv) ( ) 2
2( ( ), ) ( ( ), )
( ) ( ( ), )av a v a
a w a
.
(v) ( , )v x is continuous in ( , ) (0, ) .x
(vi) ( , )v x is entire in for each fixed (0, ).x
(vii) ( , ) ( , ), ( , ) ( , )v x v x v x v x for all ( , ) (0, ) .x
PROOF: Since (0, ) 1,w we find the value of ( , )v x at 0x from (8.87) by
analyzing ( ) ( , )av x at 0.x Using L’Hôpital’s rule
(8.89)
2( )
2 2
( ) 20 0 0 0
22
22 20
( , )lim ( , ) lim ( , ) 2 lim ( , ) lim
22 1 lim 1,
( , )
as
xa
x x x x x
x
x x
eds
s w sv x v x w x
e
xe
x w x e
So (i) is established. Similarly, from (8.87),
53
(8.90)
2 2 2( ) ( )2
( ) 2 2 2 2 2 2
4 ( , ) 2( , ) ( , ) 2 .
( , ) ( , ) ( , )
a as x sx
a
x x
e e w x ev x w x e ds ds
s w x x s w x x w x
Property (iv) follows from putting ( )x a in (8.90). For property (ii), observe that an
application of L’Hôpital’s rule gives
(8.91)
2( )2
2 20
1lim ,
( , ) 2
a sx
xx
ee ds
s w s
and thus the first term above tends to zero as 0x because 0
lim ( , ) 0.x
w x
Combining the next two terms we find using ( , ) 1w x as 0x that
(8.92)
2( )2
2
2 2
220
2( )
2 2
2 22 2
2 20
2(
2 2
0
2 ( , )( , )2
lim( , )
4 ( , ) ( , )2 2( , )
2 lim(2 2) (2 2)
4 ( , ) ( , )( , )
2 lim
as
x
x
x x
a s
x xL
x
x x x
a s
x
x
ee w x ds
s w s
w x x e
ew x w x dse e s w sx x
x e x e
ew x w x ds
s w s
)
20,
(2 2) xx e
since 0 0
lim ( , ) 1, lim ( , ) 0,x x
w x w x
and the other numerator factors above tend to 2.
54
Hence (ii) is established. Furthermore, (iii) is established directly by (8.87)-(8.88).
Summarizing, from the analysis above we thus have ,
(8.93)
( ) ( )0 0 0 0
12( ) ( )
lim ( , ) lim ( , ) 1, lim ( , ) lim ( , ) 0,
2( ( ), ) ( ( ), ) 0, ( ( ), ) ( ( ), ) ( ( ), ).( )
a ax x x x
a a
v x v x v x v x
v a v a v a v a w aa
To establish (v)-(vii), let 0 be fixed. Since ( , )w x is continuous in x and by
Theorem 3, we make use of this continuity at 0( , ) (0, ),x namely, with 1 2, there
exists 0 such that 0 x and 01( , ) 1 .
2w x It follows that for
all [0, ]x and all 0:D there cannot be a point [0, ]x with
( , ) 0.w x Accordingly, we may define a solution of (8.16) valid for all [0, ]x and
D by the reduction of order formula
(8.94)
22
2 2( , ) : 2e ( , ) .
( , )
sx
x
ev x w x ds
s w s
Moreover for all ( , ) [0, ]x D we have that ( , )v x satisfies the ICs at :x
(8.95) 2
2( , ) 0, ( , ) .
( , )v v
w
It follows from standard existence and uniqueness results (See, particularly DS [29, p.
1281, Thm. 3].) that ( , )v x extends to be uniquely-defined for (0, )x for all
.D Comparing with Definition 2, it is obvious that this extension of ( , )v x must
55
agree with ( , ).v x Moreover, since ( , )w (and also 2 ( , )w ) are known to be entire
in , it follows from standard smoothness results (See DS [29, p. 1284, Cor. 5].), that
( , )v x and ( , )v x are continuous in ( , ) [0, )x D and also that for fixed ,x
( , )v x is analytic for .D Since 0 is arbitrary, we can conclude that ( , )v x and
( , )v x are continuous in ( , ) [0, ) ,x and that ( , )v x is also entire in for fixed
.x This proves (v) and (vi). To prove (vii) it suffices to observe that for real , ( , )w x
is real-valued, which ensures that ( , )v x is real-valued in [0, ], as well as its extension
( , )v x in [0, ) so the theorem is proven.
For any 0,C
(8.96) 0
lim ( , ) ( , ) 1px
v x Cv x
and 0
lim ( , ) ( , ) 1px
v x Cv x
Corresponding to the solutions pv and v defined in (8.86) and (8.87-88) we thus have
unique choices of principal and non-principal solutions of equation (8.5) as
(8.97) 1 1
1( , ) ( , ) ( , ) ( , ),x xp py x y x xe w x xe v x
and
(8.98) 1
2( , ) : ( , ).xy x xe v x
The problem of separating ( , )v x into a linear combination of a “canonical non-
principal” solution and the above uniquely normalized principal solution ( , )pv x will
now be given. A disadvantage of the non-principal solution defined as in (8.87)-(8.88)
56
by the reduction of order formula is that it depends on the choice of the upper limit of
integration ( ).a Since the initial conditions at 0x for equation (8.16),
(8.99) 0
lim ( , ) 1x
v x
and 0
lim ( , ) 0,x
v x
fail to determine a unique solution of (8.16) by observation in (8.96), the question arises
as to how we may characterize a “canonical choice” of non-principal solution, which is
independent of the principal solution of (8.86). Since initial conditions at 0x cannot
be used to eliminate dependence of a given choice of non-principal solution on ( , ),pv x
we must find another way to characterize a “canonical” non-principal solution which
involves no dependence on .pv A similar situation occurred in the study by Atkinson
and Fulton in [7] of the equation
(8.100) ,c
u u ux
1 2,
where every principal solution has the general form, for , ,
(8.101) ( , ) 1 (1)pv x R o , 0 ,x
for some real constant .R So in this case, Fulton and Atkinson generated a canonical
choice of non-principal solution by putting
(8.102) ( , ) 1cu x higher order terms 0 x lower order terms.
That is, the zero multiplying x eliminated dependence of the canonical choice of non-
principal solution, ( , ),cu x on .pv To follow this protocol for the present problem, we
first make the following definition.
57
Definition 3: A solution npcv of equation (8.16) is a canonical non-principal solution at
0x for all of the following conditions hold:
(8.103) 0
lim ( , ) 1,npcx
v x
and
(8.104) 0
lim ( , ) 0,npcx
v x
and non-principal solution not ,npcv nonzero real constants A and B such that
(8.105) 2
( , ) ( ) ( , ) ( ) ( , ),xnp npcv x A v x B e w x
where ( , )w x is the unique principal solution from Theorems 3 and 4.
The existence of a formal power series solution of the equation (8.16), near the irregular
singular point 0x is well known and the usual ansatz,
(8.106) 1
( , ) 1 ( ) ,n
n
n
v x c x
gives rise to the 3-term recurrence relation (Compare to Luger and Neuner in [74].)
(8.107) 23
( 2)( 3),
2( 3)
k kk
k k c cc
k
1,2,3,...,k
with the intializations 1 2 30, .
6c c c
This recurrence relation gives the first few
terms of the asymptotic power series solution (8.106) as
(8.108) 3 4 5 2
6 75( , ) 1 ( ).
6 4 2 4 72
x x xv x x O x
The canonical non-principal solution of (8.16) in the above definition would necessarily
58
have the same formal power series (8.108) as its asymptotic expansion since the limiting
value in (8.103) as 0x is 1 in agreement with (8.106). For the principal solution
( , )pv x in (8.86) we have the leading term 2
xe
which tends to zero as 0x more
rapidly than all powers of .x A calculation of the asymptotic power series of this
function near 0x as 2
0
nxne a x
using the formulas from [17, p. 99] yields
2
00
lim 0x
xa e
and
12
0
0lim 0,
nix
i
in nx
e a x
ax
1.n Hence the principal solution
2
( , ) ( , )xpv x e w x
of equation (8.16) is “subdominant” to the asymptotic power
series (8.106), in the sense of Bender and Orszag, [17, p. 123]. This means that any
“large” solution of (8.16) of the form
(8.109) 2
( , ) ( , ) ( ) ( , ),xnp npcv x v x B e w x
must have the same asymptotic power series (8.108), for any ( ) 0.B Since by (8.93),
( , ) 1v x as 0 ,x thus we have, in particular,
(8.110) 2
( , ) ( , ) ( ) ( , ),xnpcv x v x B e w x
for some nonzero constant ( ).B We now define our canonical choice for the 2nd
linearly independent solution to (8.1) near 0x as
(8.111) 1
2( , ) ( , ) : ( , ).xnpc npcy x y x xe v x
59
We also observe that after multiplation of both sides of (8.110) by 1
xxe gives
(8.112) 2 1( , ) ( , ) ( ) ( , ).npcy x y x B y x
In the next section, we mirror the analysis provided in this section for 0x but for the
irregular singular point .x In doing so, we will see certain distictions develop
between the solutions defined near the irregular singular point x as compared to
those that have been defined in this section near 0.x These distinctions include but are
not limited to the finding that near the irregular singular point at ,x the controlling
factors of solutions are dependent.
60
Section 9: The 41 x Potential SL Equation and Irregular Singular Point x
Existence of a Bounded Solution to the Terminal Value Problem
The fact that x is an irregular singular point of Poincaré rank 1h for the equation
(8.1) follows from the analysis of of K. Jörgens [67], F.W. Olver [83], [20, p. 462] or
Poincaré who first gave the classification scheme in [86]. Generally speaking, this means
that near ,x solutions to (8.1) will have an asymptotic representation of the form
(9.1) 0
0
( , ) ~x n
nn
cy x e x
x
,
for some constants 0 and . Per the analysis of K. Jörgens [67], we find that
0 and 0 for the equation (8.1). We may expect that for each of the two
different choices of 0 , there will exist a linearly independent solution to ODE
(8.1) near .x In this section we prove the existence of a square-integrable solution
near x for when Im 0. To this end, we first identify two linearly independent
solutions to (8.1) near .x So as to make use of the new method of solution utilized in
Section 8, we will obtain solutions with asymptotics in the form of (9.1) by the following
method. First we make the changes of variables
(9.2) ( , ),i xy e x
and
(9.3) ( , ),i xy e x
into equation (8.1). This procedure gives two ODEs for ( , )x and ( , )x as
61
(9.4) 4
2 0,ix
and
(9.5) 4
2 0.ix
Now we employ variation of parameters for the equations
(9.6) 2 0i homogeneous equation
and
(9.7) 4
2ix
nonhomogeneous equation
to obtain the following lemma.
Lemma 3: If ( , )x is twice differentiable with respect to ,x then every solution to
(9.4) is also a solution to the Volterra integral equation
(9.8) 0
2 2 ( )
4( , ) 1 ( , )
2
x
i x i y x
x
ix E Fe e y dy
y
for some choice of the constants E and F and vice versa.
PROOF: Since 21, i xe is a fundamental system to (9.6) having Wronskian
determinant, 22 ,i xW i e variation of parameters, as applied to (9.7), yields
(9.9) 2 2
1 2( , ) 1 1 .i x i x
particular solution
x E F e u u e
It now remains to be shown that
62
(9.10) 0
2 2 ( )
1 2 41 1 ( , )
2
x
i x i y x
xparticular solution
iu u e e y dy
y
To this end, 1u and 2u are computed using the relations 11 ,
Wu
W 2
2 ,W
uW
where
(9.11)
2
2
1 42
4
01
( , ),12
i x
i x
i x
e
W e xxi e
x
2 4
4
1 01
.10
Wx
x
Integration from 0x to x of 1u and 2u as specified in (9.11) now yields
(9.12)
0 0
11 4
( ) ( , ) ,2
x x
x x
W iu y dy y dy
W y
and
(9.13)
0 0
222 4
( ) ( , ) .2
x x
i y
x x
W iu y dy e y dy
W y
Lastly we insert (9.12) and (9.13) into (9.9) and (9.8) has been established. Now let
( , )x be a twice differentiable with respect to x solution to (9.8). Computation of
( , )x and ( , )x shows that ( , )x satisfies (9.4) so the lemma is now established.
While we intend to focus our analysis mainly on our ( , )x function since
3 ( , )i xy e x is square-integrable near x for when Im 0 on account of the
exponentially decaying controlling factor ,i xe we point out here that a Volterra integral
equation for ( , )x is derived in the same manner as shown above for ( , ).x We find
( , )x satisfies the Volterra integral equation
63
(9.14) 0
2 2 ( )
4( , ) 1 ( , )
2
x
i x i x y
x
ix G He e y dy
y
for some constants G and .H The validity of relation (9.14) is bolstered by the
observation that by normalizing ( , )x with 1G and choosing 0,H application of
repeated integration by parts in (9.14) gives rise to the formal power series for ( , )x
near 0x obtained later in Section 11. Unfortunately, this same formal power series is
also obtained with H kept arbitrary so it appears that we cannot localize a unique
( , )x solution by imposing ( , ) 1x as .x On the other hand, we observe that
(9.2)-(9.3) implies that ( , )p x satisfies (9.5) where
(9.15) 2( , ) ( , ).i x
p x e x
To show that ( , )p x as defined in (9.15) indeed satisfies (9.5), we simply insert
( , )p x into (9.5) and observe that since ( , )x satisfies (9.4), ( , )p x is a solution to
(9.5). Thus the solutions to the three equations (9.4), (9.5) and (8.1) are “coupled” and it
reasons that the solution spaces for either (9.4) or (9.5) alone will be sufficient to yield
two linearly independent solutions to (8.1) via (9.2)-(9.3). This is the case since if we can
obtain one solution ( , )x to (9.4), then we will have also obtained one solution to (9.5)
by the relation (9.15) and one solution to (8.1) by (9.2). Because the solution spaces for
these equations are two dimensional, we can then utilize the reduction of order technique
so as to generate a 2nd linearly independent solution to (9.4) and thus a second linearly
independent solution to (9.5) and to (8.1) will likewise have been identified. We identify
64
here that unlike what occurs for ( , ),x ( , )x may uniquely be specified from its
Volterra integral equation (9.8) by the imposition of certain terminal conditions at .x
The subscripting of ( , )x in relation (9.15) refers to the analogue of a principal
solution however in the literature, there is no such convention towards the designation of
principal or non-principal solutions near an LP/O-N irregular singular point at .x
The question arises as to whether we can replace 0x in Lemma 3 by ; and more
particularly, whether we can take 0F and prove that there exists solutions of (9.8)
having continuity of ( , )x and ( , )x as .x To this end we first establish a
lemma.
Lemma 4: If a solution ( , )x of (9.8) that is bounded on [ , ),x 0x , then for
some 0,E ( , )x satisfies the Volterra integral equation
(9.16) 2 ( )
4( , ) 1 ( , ) ,
2
i y x
x
ix E e y dy
y
and the terminal conditions
(9.17) lim ( , )x
x E
and lim ( , ) 0,x
x
, Im 0.
PROOF: For , Im 0, , ,ii re r observe that
(9.18)
1 1 11 2 2 22 2
2 cos sin 2 cos 2 sin2 22 2 2 2 ,
iir i x ir x r x
i x ir e xe e e e e
where we have used the principal branch of . The first exponential factor above is
bounded by 1 for all x whereas the second exponential factor tends to as .x
65
Hence for ( , )x to be bounded for all x positive we must take F in (9.8) to be zero.
This gives (9.16). To verify that the terminal conditions at infinity (9.17) hold, observe
(9.19)
2
4
2
4 4
2
lim ( , ) lim 1 ( , )2
( , ) ( , )2 2
lim lim .2
i y x
x xx
i y
Lx
i xx x
ix E e y dy
y
i ie y dy x
y xE E E
ie
Hence it has been shown that provided ( , )x is bounded for all x positive and for all
in the upper half of the plane, lim ( , ) .x
x E
Observe also that
(9.20)
2
4
2
44
2
lim ( , ) lim 1 ( , )2
1 1( , ) ( , )
lim lim 0,2
i y x
x xx
i y
Lx
i xx x
d ix e y dy
dx y
e y dy xy x
ie
where again we have made use of the assumption that ( , )x is bounded, and we have
also made use of both L’Hôpital’s rule and the Liebniz rule restated below.
(9.21)
So provided that ( , )x is bounded, we have established lim ( , ) 0x
x
and the proof
of Lemma 4 is done. Now that it has been shown that a bounded solution to (9.8) must
satisfy (9.16) we follow with a theorem establishing the existence of such a bounded
solution to (9.16).
( ) ( )
( ) ( )
( , ) ( , ) ( , ( )) ( ) ( , ( )) ( )
x x
x
x x
df x y dy f x y dy f x x x f x x x
dx
66
Theorem 6: There exists a bounded solution ( , )x in
, (0, ), Im 0D x x to the Volterra integral equation
(9.22) 2 ( )
4( , ) 1 1 ( , ) ,
2
i y x
x
ix e y dy
y
such that
(9.23) lim ( , ) 1x
x
and lim ( , ) 0,x
x
, Im 0.
Moreover, for ( , ) ,x D this solution is continuous in both x and and analytic for
: Im 0.
PROOF: Let 2 ( )( , , ) 12
i y xiK x y e
so that
4
1( , ) 1 ( , , ) ( , ) .
x
x K x y y dyy
Now to construct a bounded solution to (9.22) in
,D we formulate the Neumann series solution to the Volterra Integral Equation (9.22) as
(9.24) 0
( , ) ( , )n
n
x x
,
where ( , )n x is defined by the recursive scheme for 0,1,2,3,n ,
(9.25) 1 4
1( , ) ( , , ) ( , ) ,n n
x
x K x y y dyy
and with the intialization 0 1. To prove convergence of the series (9.24), we observe
that for ,y x 0,x 0 , 0,r and ,ire we have
67
(9.26) 1
2 ( ) 2 ( )21
( , , ) 1 122
i y x i y xiK x y e r e
,
where
(9.27)
1 11 2 2
2 2 2 sin ( ) 2 cos ( )2 ( ) 2 ( ) 2 2 1 1 1.
i r y x ir y xi y x ir e y xe e e e
In (9.27), the branch cut of is taken to be on the positive real axis. Hence
(9.28) 1
2( , , ) .K x y r
From (9.28) we obtain the bounds
(9.29) 1
321 04
1 1( , ) ( , , ) ,
3x
x K x y dy r xy
(9.30) 1 6
2 14
1 1 1( , ) ( , , ) ( , ) ,
3 6x
x K x y y dy r xy
(9.31) 3
923 24
1 1 1 1( , ) ( , , ) ( , ) ,
3 6 9x
x K x y y dy r xy
and then by a mathematical induction we obtain
(9.32) 32
( , ) .3 !
nn
n n
r xx
n
Accordingly, using the bound (9.32) in (9.24) we find
(9.33) 32
0 1
( , ) ( , ) 13 !
nn
nn n
r xx x
n
.
68
Hence for 0[ , ),x x 0 0,x s.t. Im 0, and ,r R we have the majorant,
(9.34) 32
0
1
( , ) 1 .3 !
nn
nn
R xx
n
As the series of bounds (9.34) is convergent by the ratio test in 0[ , )x and ,R the
Weierstrass M-test [69, p. 413] asserts that the series (9.24) defining ( , )x is uniformly
convergent in 0[ , ) : , Im 0, .x R From uniform convergence, it
follows that ( , )x must also be continuous in x and in this domain. Since and
R are arbitrary positive numbers, we thus have that ( , )x is continuous (0, )x
and : Im 0 . Moreover, it follows from the Weierstrass Theorem that
( , )x for fixed x is analytic in the upper half plane. Substitution of (9.24) into the
integral equation (9.22) and use of the terminal condition lim ( , ) 1,x
x
for all
: Im 0, together with uniform convergence of the series (9.24) for ( , )x in
0[ , ) : , Im 0,x R proves that the series (9.24) is a solution of the
integral equation (9.22).
0x
69
Section 10: Uniqueness of the Bounded Solution
Theorem 6 in Section 9 establishes the existence of a bounded solution to the terminal
value problem (9.4), (9.23), or equivalently, the existence of a bounded solution to the
Volterra integral equation (9.22). In this section we establish that this bounded solution
is unique for all 0[ , )x x , when 0x is sufficiently large; of course, it is well known that
any constant multiple of a bounded solution defined by (9.16) will be another bounded
solution to the linear equation (9.4) yet, by selecting 1E in (9.16)-(9.17), such a
constant multiple factor is fixed to 1 and thus the solution is indeed unique. Interestingly,
it turns out that again as seen in Section 8 for the solutions near 0x , the uniqueness of
the bounded solutions near x cannot be established by the standard theorems. First
we observe that the standard Picard, as well as the Carathéodory existence and
uniqueness theorems [25, p. 3, 43] are not formulated for singularities near x but
rather they are formulated for establishing uniqueness of solutions for problems near
finite points. This would suggest that these standard theorems cannot be applied to
establish a uniqueness property for the bounded solution generated near x however it
turns out that this is not the case. We make use of the fact that the irregular singular point
at x may always be relocated to any finite t value, ,c by means of the substitution
1t c x and thus application of the existence and uniqueness analysis using the
standard Picard, as well as the Carathéodory existence and uniqueness theorems may
indeed be carried out in the t variable at the finite point t c . For instance, taking
0c in the substitution indicated above, we make the change of variables 1t x in
70
(9.4), (9.23) which relocates the singularity from x to 0t . Below we show that
despite the ability to apply these theorems near 0t , still they cannot establish a
uniqueness property for the bounded solution to the terminal value problem (9.4), (9.23).
The change of variables 1t x converts (9.4), (9.23) to
(10.1) 2
2 2( , ) ( , ) ( , ) 0,
it t t
t t
where
(10.2) 0
lim ( , ) 1t
t
and 2
0lim ( , ) 0.t
t t
Next we give equation (10.1) in the system form
(10.3)
2
1
22 1 22
0 1
( , ) ( , , ) ,2 2 2 21
gA t g ti i
gt t t t
with 0t , in the upper half of the plane, and where
(10.4) 1
2
( , )( , ) .
( , )
tt
t
The standard choice of a Lipshitz constant for which
(10.5) 0
1 2 1 20
sup ( )t t
A t L
, 0 0t ,
in some domain of g ,
(10.6) 0: (0, ] :t Z Z B
,
71
may again be taken as in [65, p. 3-4] to be
(10.7) 0 0
2,0 0, 2
2 2: 2 2 sup max 2 sup max 0,1,i
i jt t t tji jZ B Z B
g iL M
t t
.
But for the problem, this is not bounded as 0 ,t and hence fails to define a Lipshitz
constant for 0(0, ].t t Thus, in this case, the Picard theorem cannot establish
uniqueness. Similarly, in the case of the Carathéodory initial value problem at 0,t
uniqueness of the bounded solution of the IVP (10.1), (10.2) would require as in
[25, p. 43] that
(10.8) 0 0
2 1 220 0
2 2sup ( , , sup max , ( ),
t t t t
Z B Z B
ig t Z m t
t t
for some Lebesque-integrable function ( ),m t that is
(10.9) 0
0
( ) .
t
m t dt
But again this is not the case since the function 2
2 2i
t t
is not integrable near 0.t
Thus we observe that in this case, the Carathéodory existence and uniqueness theorem
likewise cannot establish uniqueness. Despite the inability of these standard theorems to
establish uniqueness of the bounded solution whose existence is guaranteed from
Theorem 6, it is indeed unique as will now be shown. To prove the uniqueness of the
bounded solution (9.24) obtained in the proof of Theorem 6, we give an argument in the
72
same style as that used in Section 8 by its reliance on the Banach fixed point theorem.
Namely, we prove the following theorem:
Theorem 7: Within the domain 0, [ , ), Im 0 ,D x x x for sufficiently large
0 0,x the terminal value problem (9.4), (9.23) has a unique bounded solution.
PROOF: Consider the Banach space
(10.10) B:= 0[ , ) : ( )C x f f x is continuous on 0[ , ) ,x
the space of complex-valued functions which are continuous on the interval 0[ , ),x with
norm
(10.11) 0
sup ( ) .x x
f f x
The metric associated with this norm on B is
(10.12) ( , ) : .d f g f g
Now consider the subset of B defined by
(10.13) S : f B : lim ( ) 1x
f x
To see that S is a closed subset of B, let nf S and f B such that ( , ) 0.nd f f
Since lim ( ) 1nx
f x
for all ,n and
(10.14) 0
lim ( ) lim ( ) sup ( ) ( ) ( , ) 0,n n nx x x x
f x f x f x f x d f f
it follows that lim ( ) 1.x
f x
Hence S is a closed set in the Banach space B and is
73
therefore itself a complete metric space with the same metric. Now consider the mapping
:T S S defined by
(10.15) 2 ( )
4 4
1 1( )( ) 1 ( , , ) ( ) 1 1 ( ) .
2
i y x
x x
iTf x K x y f y dy e f y dy
y y
To see that T maps S into S we observe that since 0[ , ),f C x Lemma 4 gives
(10.16) lim( )( ) 1.x
Tf x
To establish that (10.15) has a unique solution f S, we now prove the following claim.
Claim: For sufficiently large 0 0,x T is a contraction map on the metric space
(S , ).d To prove this claim, observe that
(10.17) 4
1( )( ) ( , , ) ( ) ( ) .
x
Tf Tg x K x y f y g y dyy
By use of the bound obtained in (9.28),
(10.18) 3
10 2
4
1( )( ) ( , , ) ,
3x
xTf Tg x f g K x y dy r f g
y
where .r It follows using the definition of the metric (10.12) on B and on S, that
(10.19) 3
10 2( , ) ( , ).3
xd Tf Tg r d f g
Hence for any fixed : Im 0, by choosing 0x sufficiently large so that
(10.20) 3
10 2 1,3
xr
74
it follows that T is a contraction mapping on S. The Banach fixed point theorem
[71, p. 300] therefore applies to S with the above metric (10.12), and so there is a unique
element f S satisfying
(10.21) 4
1( ) ( )( ) 1 ( , , ) ( ) ,
x
f x Tf x K x y f y dyy
and
(10.22) lim ( ) 1.x
f x
It remains to show that the fixed point, ( ),f x is differentiable and satisfies the additional
initial condition in (9.23),
(10.23) lim ( ) 0.x
f x
Differentiation of (10.21) gives
(10.24) 2 ( )
4 4
1 ( , , ) 1( ) ( ) ( ) ,i y x
x x
df K x yx f y dy e f y dy
dx y x y
which is continuous for all 0[ , ).x x To extablish that ( )f x also satisfies the
terminal condition (10.23) at , observe from L’Hôpital’s rule
(10.25)
22 )
44
2 ) 2 )
1 1( ) ( )
lim ( ) lim lim 0,2
i yi x
Lx
i x i xx x x
e f y dy e f xy xf xe i e
since ( ) 1f x in this limit by (10.22). This completes the proof of Theorem 7.
75
Corollary 2: The solution ( , )x in (9.22) of Theorem 6 is equal to the fixed point
( )f x in Theorem 7, and also represents the unique solution of the TVP (9.4), (9.23) on
[0, ).x
PROOF: For each fixed : Im 0, ( ) ( , )f x f x in (10.21) and ( , )x in
(9.24) are solutions of the TVP (9.4), (9.23) at least for 0[ , ).x x Hence
( , ) ( , )x f x for all 0[ , ),x x and is unique by virtue of Theorem 7. For the
unique extension of the solution ( , )x to (0, ),x for a fixed value of and a fixed
0[ , ),x x consider the IVP (10.26)-(10.27)
(10.26) 2
2 2( , ) ( , ) ( , ) 0,z x z x z x
x x
(10.27) ( , ) ( , ), ( , ) ( , ).z x w x z x w x
Standard arguments for TVPs now apply to establish uniqueness of the solution, ( , ),z x
of the TVP (10.26)-(10.27) for all (0, )x (See [93, p. 182, Thm. 9.1] or [29, Vol. II,
Thm. 3, p. 1281].). But since the solution ( , )x constructed in (9.24) also satisfies the
TVP (10.26)-(10.27), it follows from the uniqueness for this TVP that ( , ) ( , )z x x
for all (0, ).x This completes the proof of Corollary 2.
With the uniquely-defined ( , )x from Theorem 7, we take our standard square-
integrable solution to (8.5) near x to be the solution
(10.28) 3( , ) ( , ).i xy x e x
76
We now consider the equation (9.5). This equation has x as an irregular singular
point of Poincaré rank 1; however, the essential singularity exp( )i x in the y
solution (9.3) has been factored out, so that the equation, for : Im 0 will
have solutions which are bounded and continuous near .x Multiplying both sides of
(9.5) by the integrating factor integrating factor 2( ) ,i xp x e (9.5) becomes
2 2 4 0.i x i xe e x Under the changes of variables in (9.2)-(9.3) it will be
observed that solutions of the equation (9.4) and the equation (9.5) are related by
(10.29) ( , ) exp(2 ) ( , ).x i x x
Accordingly, for : Im 0, if ( , )x is taken as the “small” solution in (9.2) and
found in Theorems 6 and 7, then the “small” solution of (9.5) is uniquely-specified as
(10.30) ( , ) : exp(2 ) ( , ).p x i x x
For fixed : Im 0, there exists ( ) 0b b such that there are no zeros of
( , )x for [ ( ), ).x b We can therefore make the following definition of a “large”
solution near x for all (0, )x and all : Im 0 as follows:
77
Definition 4: For each fixed : Im 0, let ( )b be chosen as above, and define,
in terms of the “small” solution, ( , ),x a second linearly independent solution by the
reduction of order formula:
(10.31) 2 2
2
( ) 2 2
( ) ( )
( , ) 2 ( , ) 2 ( , ) .( , ) ( , )
x xi s i si x
b p
b b
e ex i x ds i e x ds
s s
for [ ( ), ).x b Here, for : Im 0, ( , )x annd ( , )p x are complex-valued
solutions of (9.4) and (9.5); but it is easy to verify that for a complex-valued solution of
(9.5), the reduction of order formula generates another complex-valued solution to (9.5).
The solution defined in (10.31) is a uniquely-defined solution of (9.5) by virtue of the
uniqueness of the solution ( , )x from Theorem 7. For each fixed : Im 0, we
can extend ( ) ( , )b x to (0, )x by choosing as initial conditions for any
( ( ), ),x b values of the above solution, ( ) ( , )b x and ( ) ( , )b x (See D.S. [29, p.
1281, Thm. 3].) We can then make a unique definition of a “large” solution of (9.5) for
(0, )x and all : Im 0, by putting for each : Im 0 :
(10.32) ( , ) :x uniquely-extended solution ( ) ( , )b x to (0, ).
We now prove the following theorem establishing properties of ( , ) :x
78
Theorem 8: The solution defined by (10.31)-(10.32) for ( , ) (0, ) : Im 0,x
satisfies the following properties:
(i) lim ( , ) 1x
x
: Im 0.
(ii) lim ( , ) 0x
x
: Im 0.
(iii) ( )( ( ), ) ( ( ), ) 0bb b : Im 0.
(iv) ( )
2( ( ), ) ( ( ), )
( ( ), )b
ib b
b
: Im 0.
(v) ( , )x is continuous in ( , ) (0, ) : Im 0 .x
(vi) ( , )x is analytic in the upper half plane for each fixed (0, ).x
PROOF: Since lim ( , ) 1,x
x
we find the value of ( , )x as x from (10.31)-
(10.32) by analyzing ( ) ( , )b x in (10.31) as .x Using L’Hôpital’s rule
(10.33)
2
2
( )
( ) 2
2 2
2
( , )lim ( , ) lim ( , ) 2 lim
( , )2 lim 1,
2
x i s
b
b i xx x x
i xL
i xx
eds
sx x i
e
e xi
i e
So (i) is established. Similarly, from (10.31), we differentiate ( ) ( , )b x to get,
(10.34) 2
2
( ) 2
( )
2( , ) 2 ( , ) .
( , ) ( , )
x i si x
b
b
e ix i e x ds
s x
Now we use definition (10.32) and L’Hôpital’s rule to obtain
79
(10.35) ( )lim ( , ) lim ( , ) 0.b
x xx x
Furthermore, since ( ( ), )b is nonzero, evaluating ( , ),x ( , )x at ( )x b and
using (10.31)-(10.32) gives
(10.36) ( ( ), ) 0, ( ( ), ) 2 ( ( ), ).b b i b
Summarizing, we thus have established properties (i)-(iv): : Im 0,
(10.36) lim ( , ) 1, lim ( , ) 0,
( ( ), ) 0, ( ( ), ) 2 ( ( ), ).
x xx x
b b i b
To prove (v) and (vi) we follow an argument similar to Theorem 5. Let 0 with
0Im 0. Since ( , )x is continuous in x and for ( , ) (0, ) : Im 0x
and lim ( , ) 1x
x
for all Im 0: Given 1 2, 0 and 0x sufficiently large
such that 0x x and 0 ( , ) 1 1 2.w x Let 0:D be in
the upper half plane. For all D there does not exist a point 0[ , )x x with
( , ) 0.x Accordingly, we may define a solution of (9.5) valid for all 0[ , )x x and
all D by the reduction of order formula
(10.37)
0
22
2( , ) : 2 ( , ) .
( , )
x i si x
x
ex i e x ds
s
Moreover, for all 0( , ) [ , )x x D we have that ( , ) satisfies the ICs at :x
(10.38) 0 0 0( , ) 0, ( , ) 2 ( , ),x x i x for all .D
It follows from standard existence and uniqueness results (See DS, [29, p. 1281, Thm.
80
3].) that extends to be uniquely-defined for all (0, )x and all .D Comparing
with the Definition 4, it is obvious that this extension of ( , ) must agree with ( , ).x
Moreover, since 0( , )x is analytic for all Im 0, and since is taken as the
branch of the square root with cut on the positive real axis (and is thus analytic for
Im 0 ), it follows from standard smoothness results (See DS [29, p. 1284, Corollary
5].), ( , )x and ( , )x are continuous in ( , ) (0, ) ,x D and that ( , )x is
analytic for .D And since 0 in the upper half plane is arbitrary, we can
conclude that ( , )x and ( , )x are continuous in ( , ) (0, ) : Im 0x and
that ( , )x is analytic in for all Im 0 for each fixed (0, ).x As (v) and (vi)
are now established, the proof is done.
We end this section by defining our standard “large” solution to (8.1) near x for
( , ) 0, : Im 0x as
(10.39) 4( , ) ( , ).i xy x e x
81
Section 11: Asymptotics For The Solutions Defined Near x
In this section, we give an asymptotic relation near x for the unique bounded
solution ( , )x to the terminal value problem (9.4), (9.23) whose existence was
established in Theorem 6 in Section 10. We also give the asymptotic relation for
3( , )y x near .x Additionally, we give the asymptotic for any second linearly
independent solution to (8.1) near .x It has already been established in this work, in
Section 10, that the terminal value problem (9.4), (9.23) has the unique bounded solution
( , )x near .x To obtain a somewhat more explicit representation for
0
( , ) ( , )n
n
x x
than as was obtained in Section 9, we carry out the Neumann series
ansatz (9.24)-(9.25) for the Volterra integral equation (9.22) with the initialization 0 1 .
We find for 0n in (9.25) that
(11.1) 2 ( )
1 4
1( , ) 1 1 .
2
i y x
x
ix e dy
y
Now applying integration by parts, so as to generate a series expansion in descending
negative powers of ,x we find the closed form representation
(11.2) 3 ( 4)
1
0
1( , ) ( 3)! 2 .
246
kk
k
ix x k i x
Next, taking 1n in (9.25) and inserting in 1( , )x as above, we likewise employ
integration by parts to find from computation of the 1st term that
82
(11.3) 6 7
2
1( , ) ( )
72x x O x
.
Iteration taking 2,3,4,n and inserting 2 3 4, , , into (9.25) then gives
(11.4) 9
3( , ) ( ),x O x 12
4( , ) ( ),x O x 15
5( , ) ( ),x O x
with the general term found to be
(11.5) 3( , ) ( ).n
n x O x
Since ( , )x as given by the Neumann series ansatz is defined as the summation from
0n to n of ( , ),n x we find the asymptotic relation near x for ( , ),x
carried out to its 6x
term, to be
(11.6) 3 4 5 6 7
3 22
1 5 1( , ) ~ 1 ( ),
4 4 726 2
i ix x x x x O x
.x
As to be expected, we observe that the asymptotic relation above tends to 1 as .x
Hence by (9.2), as ,x 3( , )y x has the asymptotic relation,
(11.7) 3 4 5 6 7
3 3 22
1 5 1( , ) ~ 1 ( ) .
4 4 726 2
i x i iy x e x x x x O x
For Im 0 and by selecting the branch of , with a branch cut on the positive real
axis, 3( , )y x can be shown to be in 2 0[ , ),L x 0 0,x the space of square-integrable
functions near infinity as a result of the exponential factor i xe
rapidly diminishing as
x for when Im 0. Now, so as to generate an asymptotic relation for a second
linearly independent solution to (8.1) near ,x we apply the formal power series
83
ansatz near the irregular singular point x to generate such a solution’s asymptotic
relation. The formal power series ansatz dates back to the work of Poincaré in [86] and
appears also in the work of Thomé [94], Birkhoff [20], Erdelyi [30], and Jörgens [67].
Thus from the relations (9.3) and (9.5) which determine a non-square integrable solution
near ,x 4 ( , ),y x we seek an asymptotic relation for ( , )x near x as a power
series in descending powers of .x Choosing the normalization from our integral equation
method where the initialization is taken to be 1, the asymptotic series representation for a
“large” solution to equation (9.5) as x is obtained by use of the formal power
series ansatz,
(11.21) 1
( , ) ~ 1 ,n
n
n
x x
1
1
( , ) ~ ,n
n
n
x n x
2
1
( , ) ~ ( 1) .n
n
n
x n n x
Inserting these asymptotic relations into (9.5) yields the three-term recurrence relation
(11.22) 23
( 2)( 3),
2 ( 3)
k kk
k k
i k
1,2,3, ,k 0 1, 1 2 0,
3
1.
6i
Iteration from this recurrence relation then gives as ,x
(11.23) 3 4 5 6 7
3 22
1 1 5( , ) ~ 1 ( ).
4 72 46 2
i ix x x x x O x
Here we point out that ( , ),x as defined in (10.31)-(10.32), must also have the same
asymptotic as given above in (11.23) because ( , ) 1x as x (Theorem 8(i)).
Also, any component of ( , )x that is linearly dependent with ( , )p x as given in
(9.15) tends to zero as x due to ( , )p x ’s rapidly decaying exponential factor.
84
Thus such a term linearly dependent with ( , )p x can have no bearing on ( , )x ’s
asymptotic relation near .x Thus (11.24)
3 4 5 6 7
3 22
1 1 5( , ) ~ 1 ( ), .
4 72 46 2
i ix x x x x O x x
Hence by (9.3), the asymptotic relation for our 4 ( , )y x in (10.39) as x is
(11.25)
So as to validate the asymptotic results, as x , obtained in this section for ( , )x
and ( , ),x equations (11.6) and (11.23), in Section 14, we’ll show by means wholly
independent of our integral equation method and the classical formal power series
method, matching asymptotic relations obtained in this section for our ( , )x and
( , )x functions. In the next section, we seek to solve a “connection problem”
pertaining to the explicitly defined solutions 1y and 2y defined near 0x and 3y
defined near .x
3 4 5 6 7
4 3 22
1 1 5( , ) ~ 1 ( ) .
4 72 46 2
i x i iy x e x x x x O x
85
Section 12: The Connection Problem Preliminaries
In this section we investigate the connection problem for equation (8.1). A connection
problem for a 2nd order linear ODE concerns the determination of linear dependence
relations that connect various forms of solution defined about different singular points.
To illustrate a connection problem, we give the following elementary example worked
out by this author.
Example: Consider the Bessel equation (2.1) in its L.-N. form on (0, ) with 1
4 ,
(12.1) 2
3 16y y y
x
,
where ire with 0 so that lies in the upper half of the complex plane.
Near 0,x the Frobenius solutions to (12.1) as given in [48] are
(12.2)
21 31 184 2 4
1 1 24 1
( 1)2 (5 / 4) ( ) 1
!(5 / 4) 2
j j j
jj j
xy x J x x
j
and
(12.3)
128 1 1
2 42 13 2
44 1
( 1)( ) 2 1 .
!(3 / 4) 22 (5 / 4)
j j j
jj j
xy x J x x
j
86
The square-integrable solution to (12.1) near x , as it appears in [95, p. 86], is now
given but with an extra 1
2x factor appearing as a result of a conversion to L.-N. form
(12.4) 1 1
(1)2 23 1 4 1 4 1 4( ) ( ) ( ) .y x H x x J x iY x
Here (1)
1 4 ( )H x is a Hankel function or Bessel function of the 3rd kind whereas
1 4 ( )Y x is a Bessel function of the 2nd kind [101] and is defined by the relation
(12.5) 1 4 1 4 1 4( ) ( ) 2 ( ).Y x J x J x
So as to connect the solutions 1,y 2 ,y and 3y by a linear dependence relation, we now
give the asymptotics for these three solutions generated near 0x and obtained from
(12.2)-(12.5) as well as the power series representations for J and J in [84, p. 217].
(12.6) 3 4 2
1 (1 ( ))y x O x , 0 ,x
(12.7) 1 4 2
2 ( 2 ( ))y x O x , 0 ,x
and
(12.8)
1 3 1 3 5 12 2 28 84 4 4 4
3 1 1 14 4 8
(1 ( )) (1 ( )) 2 (5 / 4) ( 2 ( ))
2 (5 / 4) 2 (5 / 4)
x O x x O x x O xy i
, 0 .x
87
Next we give asymptotics for 1
21 4 ( )x J x and
12
1 4 ( )x Y x near x [101, p. 199].
(12.9) 1
22 1
( ) cos ,2 4
x J x x Ox
,x
and
(12.10) 1
22 1
( ) sin ,2 4
x Y x x Ox
.x
The asymptotics for 1,y 2 ,y and 3y near x from (12.2)-(12.5), (12.9)-(12.10) are
(12.11) 3
1 2 3 841 2 (5 / 4) cos( 3 8) 1y x O x , x ,
(12.12)
3 11
84 22 2 1 (5 4) sin 3 8 cos 3 8 1 ,y x x O x
,x
and
(12.13)
1 1
1 2 1 4 1 2 1 42 23 2 cos 3 8 2 sin 3 8 1 ,y x i x O x .x
88
Finally, by use of the asymptotic relations (12.6)-(12.8), (12.11)-(12.13) worked out
above, we solve an associated connection problem by identifying the connection
coefficients ( )A and ( )B by means of limit operations where
(12.14) 3 1 2( , ) ( ) ( , ) ( ) ( , ).y x A y x B y x
Taking the limit on both sides of (12.14) as 0x yields
(12.15)
54
1 8
2 (5 4)( ) .B i
Whereas taking the limit on both sides of (12.14) as x yields
(12.16) 1 8 1 8
5 54 4
( ) .2 (5 4) 2 (5 4)
A i
89
As both ( )A and ( )B in the linear dependence relation (12.14) have been obtained, this
connection problem has been solved. This Bessel equation example serves to
demonstrate what can be considered as a general method to solve connection problems
for 2nd order SL equations. Namely, for a 2nd order SL equation, the knowledge of (6)
select asymptotic relations in total for (3) select solutions is sufficient to solve a
connection problem. We mention here that the ease of application of this (6) asymptotics
method towards the Bessel equation connection problem stems from the great amount of
knowledge that has been amassed on Bessel functions of the 1st, 2nd, and 3rd kinds and
particularly the asymptotic relations for 1,y 2 ,y and 3y being well-known near BOTH of
the singular endpoints, 0x and x (See [84, p. 217], [101, p. 199].). Whereas for
general 2nd order SL equations, the determination of (6) select asymptotic relations for (3)
select solutions that do not invoke known special functions, their identitites, &
asymptotics, like those that were utilized in the Bessel equation problem, can be
characterized as the primary obstacle in the application of this (6) asymptotics method to
solve connection problems for 2nd order SL equations. Now returning to our 41 x
potential SL equation (8.1) with ( , )w x and ( , )npcv x as given in Theorem 4, Section 8,
and in equation (8.110),
1 2
1 1
( , ) ( , )
( , ) , ( , )x x
npc
y x y x
xe w x xe v x
forms a specific fundamental
system to equation (8.1) containing solutions generated about the irregular singular point
0.x As (8.1) is a linear ODE, 3( , )y x and 4 ( , )y x , the solutions formulated near
90
x and defined by the equations (10.28) and (10.39), must necessarily be linear
combinations of 1( , )y x and 2 ( , )y x . That is,
(12.17) 3 1 1 2 2( , ) ( ) ( , ) ( ) ( , ),y x C y x C y x (0, ),x Im 0,
and
(12.18) 4 1 1 2 2( , ) ( ) ( , ) ( ) ( , ),y x D y x D y x (0, ),x Im 0.
To solve these above connection problems, the connection coefficients 1( )C and 2 ( )C
or 1( )D and 2 ( )D must be determined. We for now focus on the connection problem in
relation (12.17). To this end and in pursuit of the connection coefficients 1( )C and
2 ( ),C we observe here that thus far only (3) of the needed (6) asymptotic relations for
the solutions 1y , 2y , and 3y have been obtained in the body of this work. Namely, in
Sections 8 and 11, utilizing Volterra integral equations of the 2nd kind as well as formal
power series, we have obtained the rather explicit asymptotics
(12.19)
3 4 5 21
6 7
1
5( , ) ~ 1 ( ) , 0 ,
6 4 2 4 72x
x x xy x xe x O x x
(12.20)
3 4 5 21
6 7
2
5( , ) ~ 1 ( ) , 0 ,
6 4 2 4 72x
x x xy x xe x O x x
and as ,x
91
(12.21)
3 4 5 6 7
3 3 22
1 1 5( , ) ~ 1 ( )
4 72 46 2
i x i iy x e x x x x O x
.
However, we have yet to identify the asymptotic behaviors of these three solutions near
the singular endpoints for which these solutions were not defined about. That is, we still
lack the asymptotic behavior of 1y and 2y near x as well as the asymptotic behavior
of 3y near 0.x Moreover, the integral equation technique that we’ve developed and
utilized in this dissertation does not seem to apply so as to yield the (3) missing
asymptotics. Nonetheless, we may still proceed with only a (3) asymptotics approach
and pass the limit as x tends to zero in (12.17) to find the weak asymptotic relations
(12.22) 3 1 1 2 2
0 0 0lim ( , ) ( ) lim ( , ) ( ) lim ( , )x x x
y x C y x C y x
1 1 1
1 2 2( ) ( ) ( ) ,x x xC O xe C O xe C O xe
0 .x
Thus from (12.22), near 0,x we have found that for 0 0,x the 2 0[ , )L x solution
defined near ,x 3( , ),y x inherits the asymptotic behavior of the dominant non-
square-integrable solution 2 ( , )y x that received its definition near 0.x That is,
(12.23) 1
3 2( , ) ( ) 1 (1) , 0 .xy x xe C o x
For similar reasons, because both 1( , )y x and 2 ( , )y x are necessarily linear
92
combinations of 3( , )y x and 4 ( , )y x as defined in (10.28), (10.39), we may also expect
that the solutions 1( , )y x and 2 ( , )y x likewise inherit the asymptotic behavior of the
dominant solution 4 ( , )y x as x . That is,
(12.24) 1 1( , ) ( ) 1 (1) , ,i xy x K e o x
and
(12.25) 2 2( , ) ( ) 1 (1) ,i xy x K e o x .
where 1( )K and 2 ( )K are yet unknown constants dependent on . Here we point out
that the asymptotic relations in (12.23)-(12.25) are not as explicit as those given in
(12.19)-(12.21) due to the presence of the still unknown lambda-dependent constants
2 ( )C , 1( )K , and 2 ( )K appearing in them which render them less useful. This weak
nature of the asymptotics (12.23)-(12.25) is a result of them being obtained not by use of
integral equations like those that furnished the explicit asymptotic relations (12.19)-
(12.21), but rather by linear dependence arguments. For this reason, we now provide
some further justification of the weak asymptotic assertions in (12.23)-(12.25) by
applying the method of dominant balance as given in [17] to confirm the respective
controlling factors for 3( , )y x as 0 ,x and for 1( , )y x & 2 ( , )y x as .x For
instance, to show that indeed 1
3( , ) , 0 ,xy x O xe x as stated in (12.23), we
substitute into equation (9.4)
93
(12.26) ( ) ( ), ( ),S x S xe e S x and 2( ) ( ) ( )S xe S x S x
to obtain
(12.27) 2
4
1( ) ( ) 2 0.S x S x i S
x
Now assuming that 2
( ) ( ) , 0 ,S x S x x an assumption that will be shown to be
consistent in retrospect, a “dominant balance” is struck between 2
( )S x and 4
1
x as
0x yielding the asymptotic differential equation
(12.28) 2
4
1( ) ~ , 0 .S x x
x
Now, per the method of dominant balance, replacing the “is asymptotic to” symbol above
by equality and solving yields
(12.29) 1
( )S xx
.
As a check, this is consistent with the assumption that 2
( ) ( ) , 0 ,S x S x x since
23
4
12 , ,S x S
x
and hence 2S o S . We improve the controlling factor
estimate by replacing ( )S x by 1
( )A xx
assuming that 1
( )A xx
, 0 ,x an
assumption that may also be shown consistent in retrospect. Computation of ( )S x and
( )S x and insertion into (12.26) yields
(12.30) 2 3
2 2
2 1( ) ( ) 2 ( ) 2 2 ( ) 0.A x A x x A x i i A x
x x
94
We assume further that 2 3
2 2
1 2, , 2 , 2 , 2A A i i A A x
x x , all of which,
may be shown to be consistent assumptions in retrospect after carrying forth this analysis.
Thus by the assumptions above, a dominant balance is struck between the relatively large
terms in (12.30), 2
2A
x and
32x , as 0x , thus giving the asymptotic differential
equation
(12.31) 1
( ) ~ , 0 .A x xx
Solving as before then yields ( ) ln .A x x We can now observe that this representation of
( )A x is consistent with all of the above made assumptions. Hence we may now conclude
that the controlling factor for ( , )x is 1 ln x
xe
or 1
xxe where we have rejected the
minus sign as ( , )x must necessarily blow up as 0x as determined from equations
(12.17)-(12.19) provided that 2 ( ) 0.C This computation thus supports the validity of
the assertion in (12.23). We will now likewise employ the method of dominant balance
to confirm that the solutions to (8.1) defined near 0,x 1( , )y x and 2 ( , ),y x indeed
have the controlling factor ,i xe as ,x which blows up in this limit for when
Im 0. To this end, into equations (8.15) and (8.16), we substitute respectively
(12.32) ( )B xw e and
( )C xv e ,
and insert the corresponding derivatives into equations to obtain
95
(12.33) 2
2
2 2( ) ( ) ( ) 0,B x B x B x
x x
and
(12.34) 2
2
2 2( ) ( ) ( ) 0.C x C x C x
x x
Under the assumptions that 2
( ) ( )B x B x and 2
( ) ( ) ,C x C x ,x we find
(12.35) 2 2
( ) , ( ) ~ .B x C x
Hence per the method of dominant balance we find
(12.36) ( ), ( )B x C x i x ,
which gives the expected controlling factor for both 1( , )y x and 2 ( , )y x as ,i xe
,x where we have rejected the plus sign since 1( , )y x and 2 ( , )y x must
necessarily blow up near x by the requirement that they are both as still
undetermined linear combinations of 3( , )y x and 4 ( , )y x defined by the equations
(9.2)-(9.3) and where we have assumed that the linear dependence relations for 1( , )y x
and 2 ( , )y x contain nonzero 4 ( , )y x coefficients. That is, we have rejected the plus
sign above since for any fixed with Im 0, i xe as x whereas
0i xe as .x Thus the validity of relations (12.24)-(12.25) have likewise been
bolstered using the method of dominant balance. Despite the method of dominant
balance acting to bolster the validity of the weak asymptotic relations (12.23)-(12.25), it
96
unfortunately did not identify the constants 2 ( ),C 1( ),K and 2 ( )K that appear in
them. We now proceed further to address the connection problem (12.17). By
employing the weak asymptotics (12.23)-(12.24) in (12.17) and letting x pass to ,
(12.37) 3 1 1 2 2lim ( , ) ( ) lim ( , ) ( ) lim ( , )
x x xy x C y x C y x
or
(12.38) 1 1 2 20 ( ) ( ) ( ) ( ).C K C K
Furthermore, we find from (12.19)-(12.23) that
(12.39) 32
02
( , )( ) lim .
( , )x
y xC
y x
Interestingly, this somewhat general connection coefficient formula, (12.39), formulated
specifically in this analysis of the 41 x potential SL equation, may be expected to apply
to a broader class of 2nd order SL equations with two LP singular endpoints. This is due
to the nature of the fundamental systems defined about LP singular endpoints where for
each fixed , there exists one dominant non-square integrable solution. For
instance, as the Bessel equation (12.1) is 2nd order with two LP singular endpoints at
0x and ,x this equation fits the description above and so we may expect that the
connection coefficient, ( ),B given in (12.15), could be calculated by use of relation
(12.39) as 2 ( )C using the asymptotic relations (12.7)-(12.8). We confirm that this is
indeed the case. Namely, ( ),B as given in (12.15), calculates directly as 2 ( )C using
relation (12.39) and asymptotic relations (12.7)-(12.8) thus affirming that relation (12.39)
97
has a broader applicability towards 2nd order SL equations with LP singular endpoints
other than equation (8.1). Carrying on with our investigation we find from (12.25) that
(12.40) 3 21
02 1
( , ) ( , )( ) lim lim .
( , ) ( , )xx
y x y xC
y x y x
Now, so as to pursue the limits in (12.39) and (12.40) and thus solve our connection
problem, we try use of the asymptotic (12.20) as well as the solution representation for
3( , )y x in (9.2) to give
(12.41) 2 10
( , )( ) lim
( , )x xnpc
xC
xe v x
.
Likewise, using our weak asymptotics for 1( , )y x and 2 ( , )y x near x given by
(12.24)-(12.25), we find that
(12.42) 21 21
01
( , ) ( )( , )( ) lim lim ( ) ,
( , ) ( )( , )
npc
xx xnpc
v x KxC C
w x Kxe v x
where 1( )K and 2 ( )K are the respective proportionality constants in the relations
(12.43) 1
i xy e and 2
i xy e for Im 0 and as .x
We can now re-emphasize the chief difficulty in pinning down the connection
coefficients 2 ( )C and 1( ),C namely, while we know that the unique ( , )x defined
near x in (12.41) must blow up as 0x at a rate proportional to that of the blow
up rate for 1
xxe under this same limit, it still yet remains unclear as to how to identify the
98
proportionality constant 2 ( )C appearing in (12.23) and thus the explicit asymptotic
behavior of ( , )x (defined near x ) near the other singular endpoint 0x as is
needed to solve the connection problem (12.17). Likewise, while it is clear from (12.43)
that both 1y and 2y must blow up at rates proportional to i xe
as x , it is likewise
still unclear as to how to obtain the proportionality constants 1( )K and 2 ( )K that
would yield the explicit asymptotics additionally needed to solve the connection problem
(12.17). In the next section, we will summarize the work of W. Bühring towards a
differential equation that is inherently connected to our own. Though the methods that he
utilized are very different from our own, this inherent connection will enable a pathway
forward.
99
Section 13: A Related Investigation of W. Bühring’s
In this section, we give a synopsis of the work of W. Bühring from his 1974 J.M.P. paper
[23] and mention in passing that the analysis in [23] is also quite similar to that which
appears in the 1993 paper [24] by this same author. We shall restrict our attention to the
following equation used [23, p. 1453, eq. 27],
(13.1) 2
2 2 2 2 21( ) ( ) ( ) 0.2
x g x xg x x k x g x
There exists a strong symmetry relation between the solutions to (13.1) defined near
0x and those defined near x which enables connection formulas for these
solutions to be obtained explicitly. Near 0x , the multiplicative solution ansatz
(13.2) 2
2( ) , , \ 0 ,n
n
n
n
g x d x x
is used. Taking the 0n constants 0 0 1,d d the coefficients in the above
convergent Laurent expansions satisfy the recurrence relation
(13.3) 2
2 2 2
2 2 2 2 2
12 0,
2n n nn d d k d
, 2, 1,0,1,2,n .
This three-term recurrence relation can be written as an infinite system of equations in the
unknowns 2 , 0, 1, 2,...nd n and the system is solvable when the determinant of the
infinite matrix is zero; such determinants arise in connection with Mathieu and modified
Mathieu equations and are called “Hill” determinants. It is well-known that there exists
two solutions from setting the Hill determinant to zero and many different
100
approximate methods exist to find . See for instance [24], [55], [80], [87]-[88],
and [79]. The strong symmetry which connects behavior of the solutions to (13.1) near
0x and x arises from the fact that the above coefficients in the Laurent expansion
(13.6) satisfy the property
(13.4) 2 2
2 2 .n
n nd k d
From this symmetry in the coefficients, it follows that for all 0, ,x
(13.5) exp ,2
i k ig k g x
x
and same for replacing . The key to obtaining relations of linear dependence for
solutions defined at 0x and x is to represent them as generalized Laplace integrals
(See the work of Horn [57]-[64], Turritin [97], or Knoblock [70] for some of the earlier
work on Laplace integrals.) of the form,
(13.6) ( ) exp ( ) ,2
C
xg x xt v t dt
i
where C is a suitable contour in the complex t plane and ( )v t satisfies a 4th order
equation. The extra x factor in (13.6) is introduced by Bühring to allow greater
flexibility and with a view to enable easier numerical calculations for using
approximation methods. In [66, p. 186-189, 439-444], E.L. Ince details the theory of
generalized Laplace integral solutions. We state here the main result as a Theorem.
101
Theorem 9: Let 0 0
n ms r
x rs x
r s
L a x D
be an thn order differential operator with
polynomial coefficients of degree at most .m Let 0 0
:n m
r s
t rs t
r s
M a t D
be the
corresponding thm order differential operator and let the “bilinear concomitant” be the
series of terms of the form ( )s
xt r
s
de t v t
dt
for 0,1,2,..., 1,s m 0,1,2,..., .r n
If ( )v t is a solution to the adjoint equation ( ) 0tM v t and contour C is chosen so that
the bilinear concomitant has the same value at both termini of ,C then the generalized
Laplace integral ( ) ( )xt
C
W x e v t dt is a solution to [ ] 0.xL W
We refer the reader to [66, p. 438-444] for further details on this theory and for the proof
of Theorem 9. We will now restrict our attention to the case that occurs in Bühring’s
investigation, namely, the case where the polynomial coefficient function of highest
degree m is also the coefficient function for the highest order derivative term in the
differential operator [ ]xL and give here a brief sketch of Ince’s method for the proof of
Theorem 9 in this case. Let the differential operator [ ]xL be given by
(13.7) 1
0 1 11[ ] : ( ) ( ) ( ) ( ) [ ],
n n
x n nn n
d d dL P x P x P x P x
dx dx dx
where ra is the leading coefficient of polynomial ( ),rP x 00,1,2, , , ( 0).r n a
It can now be shown that application of the operator [ ]xL to the Laplace integral
( ) ( )xt
C
W x e v t dt followed by repeated integration by parts gives (See [66, p. 443].)
102
(13.8) 1 1
0 1 1
( ( )) ( ( ))[ ( )] ( , ( )) ( 1) ,
m n m nxt m xt
x m m
C
d t v t d t v tL W x B e v t a a e dt
dt dt
where ( , ( ))xtB e v t is the bilinear concomitant arising from the Laplace integral ( )W x and
is given as in [106, p. 149, eq. 35.3] by the relation
(13.9) 1 1
1
0
( , ( )) ( 1) ( ) ( ( ) ( )) ,n n
xt m k xt n m m k
k
k m k
B e v t e b t v t
where ( ), 0,1,2, ,kb t k m are the polynomial coefficient functions that appear in
(13.10) 1
0 1 1[ ] : ( ) ( ) ( ) [ ].
m m
t mm m
d dM b t b t b t
dt dt
From the integral in (13.8), it follows that ( )v t satisfies the adjoint equation [66, p. 123]
(13.11)
( ) ( 1)1
0 1 1[ ( )]: ( 1) ( ) ( ) ( 1) ( ) ( ) ( ) ( ) ( ) ( ) 0.m mm m
t m mM v t b t v t b t v t b t v t b t v t
As Bühring does not give much detail in [23] for the origination of his ( )v t equation [23,
p. 1453, eq. 33], we now apply Theorem 9 and show its determination. From the
equation (13.1), we have
(13.12) 2
4 3 4 2 2 21( ) ( ) ( ) 0.2
x g x x g x x x k g x
Letting ( ) ( ),2
xg x W x
i
we substitute into (13.12) to obtain the ( )W x equation,
(13.13) 24 3 4 2 2 2 2( ) (2 1) ( ) 1 2 ( ) 0.x W x x W x x x k W x
From (13.13), Theorem 8 defines the differential operator,
103
(13.14) 4 3 24 2
22 2 2 2
4 3 20 0
1 (2 1) 1 2 .s
r
t rs sr s
d d d dM a t t t k
dt dt dt dt
It then follows that for any generalized Laplace integral to satisfy (13.13), ( )v t must
satisfy the adjoint operator differential equation ( ) 0tM v t or specifically,
(13.15)
4 3 222 2 2 2
4 3 2
2 (4) (3) 2 2
( ) 1 ( ) (2 1) ( ) 1 2 ( ) ( )
7 51 ( ) 7 2 ( ) ( ) 0.2 2
t
d d dM v t t v t t v t v t k v t
dt dt dt
t v t tv v t k v t
This relation above is [23, p. 1453, eq. 33]. Per Theorem 9, contour C in (13.6) is
chosen to make the bilinear concomitant associated with this transformation,
(13.16)
2 3 2 2
2
( 1) 2 2 3 2 3
( , ( )) exp( ) .1 1(2 1) 2 ( )
2 2
xt
t x v x v xv v t x v xv v xv v
B e v t xttx v txv tv xv v xv v
the same at both ends of C and such contours are not unique. Equation (13.15) has RSPs
at t i and one irregular singular point at .t The domain of validity for the four
Frobenius solutions to (13.15) defined near t i and the four Frobenius solutions
defined near t i overlap inside the unit disk : 0 1 .t t Furthermore, the four
indicial roots from the Frobenius theory, near the regular singular points, ,t i are
identically 0,1,2, 1 2 , 1
2 1 .2
n Outside the the unit disk in the t plane,
104
1 ,t t convergent Laurent series solutions for are
(13.17) 1 2
2( ) ,n
n
n
n
v t t b t
where 2nb satisfies for 0, 1, 2, ,n
(13.18)
22
2
2 2
2 2 2 2
12 2 1 2 2
2
2 1 2 2 1 2 2 0.
n
n n
n n n b
n n n n b k b
An exact relationship holds between the Laurent series expansion coefficients in
equations (13.2) and (13.17) as (Compare the recurrence relations (13.3) and (13.18).),
(13.19) 2 2
( 1 2 ), .
( 1)m m
mb d
These ( )v t are in turn used to define the generalized Laplace integral solutions to (13.1),
(13.20)
3
1( ) ( 1) exp( ) ( ) , arg , .
2 22C
g x x xt v t dt xi
The four Frobenius solutions to (13.15) near t i are
(13.21) 1 2
0
( ) 1 1 2 1 1 ,n
n
n
v t ti A ti F ti
and
(13.22) 0
( ) 1 1 1 , 0,1,2,j n
j n j
n
u t ti A j ti G ti j
whereas the four Frobenius solutions near t i are
105
(13.23) 1 2
0
( ) 1 1 2 1 1 ,n
n
n
v t ti A ti F ti
and
(13.24) 0
( ) 1 1 1 , 0,1,2.j n
j n j
n
u t ti A j ti G ti j
The radius of convergence for (13.21)-(13.22) is 2t i whereas the radius of
convergence for (13.23)-(13.24) is 2t i . All of the Frobenius solutions to (13.15)
have coefficients that satisfy a three-term recurrence relation containing dependence on
indicial root q and these coefficients, ,nA q for 0,1,2, 1 2,q are the same for the
Frobenius solutions near both t i This recurrence relation [23, p. 1455, eq. 49] is
(13.25)
1
2 2
3
1 2 1 2( ) ( )
2 1 2
( ),2 1 2 1 2
n n
n
q n q nA q A q
q n q n
kA q
q n q n q n q n
0 ( ) 1A q for 0,1,2, 1 2,q 1(0) 0,A 2(0) 0,A 1(1) 0;A 0n
if 2, 1 2;q 1n if 1;q 2n if 0;q 1 2( ) ( ) 0.A q A q
Because the eight Frobenius solutions to (13.10), ,v 0 ,u 1 ,u 2 ,u ,v
0 ,u 1 ,u and 2 ,u
are all convergent in the common domain : 1t t , there necessarily exists a
relation of linear dependence for vin terms of ,v
0 ,u 1 ,u and 2 ,u such that
106
(13.26) 2
0
( ) ( ) ( ),j j
j
v t Ev t B u t
and where the connection coefficients 1 2, , ,E B B and 3B depend on the parameters
, , , and k . Furthermore, formulas to determine these connection coefficients
1 2, , ,E B B and 3B are obtained in [23] by evaluating (13.26) and its first three
derivatives at 0,t a t value inside the circle of convergence for all of ,v ,v
0 ,u
1 ,u and 2 .u It thus follows that these connection coefficients are solutions of the linear
system,
(13.27)
0 1 2
0 1 2 0
0 1 2 1
0 1 2 2
(1) (1) (1) (1) (1)
(1) (1) (1) (1) (1)
(1) (1) (1) (1) (1)
(1) (1) (1) (1) (1)
F G G G E F
F G G G B F
F G G G B F
F G G G B F
.
From (13.26), the linear combination,
(13.28) 2
0
( ) ( ), ( ) (1 ),j j
j
G z B G z u t G it
and ( ) (1 ),u t G it
is introduced along with the connection relations,
(13.29) ( ) ( ) ( ),v t Ev t u t
and
(13.30) 2( ) 1 ( ) ( ).u t E v t Eu t
Using the relations (13.29)-(13.30) which connect, in the common domain of
convergence, the Frobenius solutions v and u
defined near t i to the corresponding
107
solutions v and u
defined near ,t i another solution to equation (13.15) is,
(13.31) 2( ) ( ) ( ) 1 ( ) ( ),w t v t u t E E v t E u t
and this solution is defined in the common domain of convergence of the four Frobenius
solutions about t i and t i . An analytic continuation argument on a contour
contained in the common domain of convergence yields [23, p. 1454-1455, eq. 72],
(13.32)
( ) ( ), 2
( ) cos( )exp ( 1) exp 2 ( 1 2) ( ) ( ) , 2,
cos( )
v t u t t i
w t Ei i v t u t t i
where the normalization is
(13.33) exp ( 1 2) sin(( ) )
:cos( )
i
,
and exp 2 ( 1) ( 1 2)cos( )
:sin( ( ))
i
.
Analytical continuation of ( )w t yields that ( )w t is proportional to the Laurent series
solution (13.18). Finally the following formula connecting the four Frobenius solutions
near t i , and the four Frobenius solutions near ,t i and ( )v t to ( )w t is
108
(13.34)
where the constant of proportionality is
(13.35)
1 2
(1 )/21
(1 2 ) ( ) sin( ( ))( 1) ( 1) cos( ) ( 1):
2 ( 4) (2 )
F G
v i
.
To obtain this constant of proportionality , equate the two ( )w t representations in
the overlapping domains 2t i and 1,t solve for and evaluate at any t value
in the common domain of convergence. In particular, Bühring chooses to evaluate at
2 .t i Finally, we give the means to connect solutions to the ( )g x equation (13.1). Two
new generalized Laplace Integral solutions near x to equation (13.1) are defined by
taking suitable choices of solutions of the 4th order ( )v t ODE as well as contours in the
t plane. These two solutions are
(13.36)
1
(1) 1( ) exp( ) ( ) , 2 arg 2,
2C
g x x xt w t dt xi
and
(13.37)
2
(2) 1( ) exp( ) ( ) , 2 arg 2,
2C
g x x xt w t dt xi
where
cos( )(1 2 )exp ( 2)( 1) exp ( 1 2) ( ) ( ) , 2
sin( ( ))
cos( )( ) (1 2 )exp ( 2)( 1) exp ( 1 2) ( ) ( ) , 2
sin( ( ))
( 1) ( ), 1,
i i v t u t t i
w t i i v t u t t i
v t t
109
(13.38) ( ) : (1 2 )exp[ ( 2) ] ( ), 2,w t i v t t i
and
(13.39) ( ) : (1 2 )exp 2 ( ), 2,w t i v t t i
and where 1C is a contour from (1 ),t i encircling the RSP at t i , and then
returning to (1 )t i and where 2C is a contour from (1 ),t i encircling
the RSP at ,t i and then returning to (1 ).t i For the contours 1C and 2 ,C
is a fixed real number and is less than 1 2. The solutions (13.36)-(13.37) have known
behavior as x given by relations [23, p. 1457, eq. 81a, 81b]. The means to extract
asymptotic relations from generalized Laplace integrals can be found in Ince [66, p. 444].
Near 0,x two additional solutions to equation (13.1) and of the form given in (13.16)
are defined. Here the solutions of the ( )v t equation used are,
(13.40) 1 2
2( ) ( 1) ( ) ( 1) ,n
n
n
n
v t v t t b t
, 1,t
and the contour 3C from ,t i encloses both RSPs ,t i and returns to
t i with as before and 1t 3t C . These solutions thus have the form
(13.41)
3
1( ) ( 1) exp( ) ( ) , , 2 arg 2.
2C
g x x xt v t dt xi
Here, substitution of the Laurent expansion for ( ),v t (13.17), into (13.41) and
integrating term-wise will confirm that 2
2( ) ,n
n
n
n
g x d x
i.e. that ( )g x are the two
110
solutions, for , defined by the Laurent expansion (13.2). We now state as a
theorem Bühring’s connection relations between and
Theorem 10: If (13.6), (13.34), (13.36)-(13.39), and (13.41) hold, then the solutions to
(13.1): ( ),g x ( ),g x (1) ( ),g x and
(2) ( )g x satisfy the linear connection relations,
(i) (1) (2)( ) exp 2 ( ) exp 2 ( ), ,g x i g x i g x
(ii) (1) 1( ) exp ( 2) ( ) exp ( 2) ( ) ,
2 sing x i g x i g x
i v
and
(iii) (2) 1( ) exp ( 2) ( ) exp ( 2) ( ) .
2 sing x i g x i g x
i v
PROOF: From the generalized Laplace integral (13.6), we insert ( )w t in for ( )v t and
choose contour 3C . We compare with (13.41) and since 1t 3t C , we have
( ) ( 1) ( )w t v t as given in relation (13.34) so the LHS is ( ).g x Now on
the RHS, we apply a deformation of the contour argument [26]. First we define the
contour 3 2 1C C C , where 1 1, 0,t t and with suitable contours as follows
2 :C From ,t i rounding the RSP ,t i and returning to 1 1 ,t t i
: The vertical line segment in the t plane from 1 1t t i to 1 1 ,t t i
1 :C From 1 1t t i rounding the RSP ,t i and returning to (1 )t i .
See the figure below.
( ),g x ( ),g x
(1) ( ),g x (2) ( ).g x
111
Figure 1: Suitable Contours I
1C 3C 3C
2C
As such deformations of contours in regions where the integrand is analytic do not effect
the integral result [26, p. 65], we replace contour 3C with 3C giving the relation
(13.42)
3
2
1
( ) ( 1) exp( ) ( )2
exp( ) ( )2
exp( ) ( )2
exp( ) ( ) ,2
C
C
C
xg x xt v t dt
i
xxt w t dt
i
xxt w t dt
i
xxt w t dt
i
and then pass 1t to . In this limit, only the 1st and 3rd terms above contribute to
integral. To see this, we parametrize as ( ) :t 1 2 2, 1 1 ,t t it t so that
(13.43) 1 1
1
1 2 2
1
lim exp( ) ( ) lim exp( )exp( ) ( ) .t t
xt v t dt xt ixt v t dt
Bounding the integral on the RHS we find that
112
(13.44)
1 1
1 2 2 1 2 2
1 1
1 1
1 2 2 1 2
1 1
exp( )exp( ) ( ) exp( ) exp( ) ( )
exp Re exp Im ( ) exp Re ( ) ,
xt ixt v t dt xt ixt v t dt
x t x t v t dt K x t v t dt
for some scalar .K Observe that with the restriction 2 arg 2,x the exponential
factor 1exp Re 0x t as 1 .t Moreover, in the limit as ,t the four linearly
independent solutions to (13.15) behave asymptotically as 1 1 2exp 2 ,t kt
, 1, , 1,i i (See [23, p. 1456].). This can be checked by writing (13.15) with the
change of variable 2 ,t s and then applying a standard theorem on formal asymptotic
solutions in the complex s and t planes ([25, ch. 5, Theorem. 2.1]). It follows that the
kernel 1exp Re x t dominates as 1t in (13.44). Thus we have shown that
(13.45) 1
lim exp( ) ( ) 0t
xt v t dt
as 1 ,t x s.t. Re[ ] 0.x
So (13.45) establishes that the contour integration over will not contribute as we pass
1 .t From relations (13.36)-(13.37), in this limit, we will now show the RHS of
(13.42) is a linear combination of (1) ( )g x and
(2) ( ).g x Observe from (13.36)-(13.39),
and (13.45),
113
(13.46)
3
1
1 2
1 2
( ) exp( ) ( )2
lim exp( ) ( ) exp( ) ( )2 2
exp( ) ( ) exp( ) ( ) .2 2
C
tC C
C C
xg x xt w t dt
i
x xxt w t dt xt w t dt
i i
x xxt w t dt xt w t dt
i i
Next we deform contours 1C and 2C so that they approach and depart from the
singularities that they enclose along parallel coinciding rays at the same latitude as the
singularities that they enclose and feature counter-clockwise radius circles, 1t i
C
and
2 ,t i
C
around the singularities t i and t i respectively. The contour integration
along the oppositely-oriented coinciding rays cancel. For 2, we make use of (13.34)
to obtain
(13.47)
1 2
1 2
1
exp( ) ( ) exp( ) ( )2 2
exp( ) ( ) exp( ) ( )2 2
cos( )exp( ) (1 2 )exp ( 2)( 1) exp ( 1 2) ( ) ( )
2 sin( ( ))
t i t i
t i
C C
C C
C
x xxt w t dt xt w t dt
i i
x xxt w t dt xt w t dt
i i
xxt i i v t u t
i
2
cos( )exp( ) (1 2 )exp ( 2)( 1) exp ( 1 2) ( ) ( ) .
2 sin( ( ))t i
C
dt
xxt i i v t u t dt
i
Two of the above contour integrals are zero by the Cauchy-Goursat Theorem; namely,
the contour integrals with integrands containing ( )u t and ( )u t
as the integrands for
114
these contour integrals are analytic in the vicinity of the singularities t i and t i
respectively. The contour integrals containing ( )v t and ( )v t
remain so the RHS is
(13.48)
1 2
1
2
exp( ) ( ) exp( ) ( )2 2
( )exp (1 2 ) exp( ) ( )
2 2
exp (1 2 ) exp( ) ( ) .2 2
C C
C
C
x xxt w t dt xt w t dt
i i
i xxt v t dt
i
i xxt v t dt
i
Lastly we invoke relations (13.38)-(13.39) to obtain
(13.49)
1 2
1 2
(1) (2)
exp( ) ( ) exp( ) ( )2 2
exp exp( ) ( ) exp exp( ) ( )2 2 2 2
exp ( ) exp ( ),2 2
C C
C C
x xxt w t dt xt w t dt
i i
i x i xxt w t dt xt w t dt
i i
i ig x g x
so (i) is established. To see that (i) (ii) and (iii), we write relation (i) for in
the form
(13.50)
(1)
(2)
1 exp 2 1 exp 2( ) ( )
( ) ( )1 exp 2 1 exp 2
i ig x g x
g x g xi i
.
Assuming that the characteristic index is not an integer, the coefficient matrix above is
nonsingular and we can solve the system to obtain
(13.51)
(1)
(2)
exp 2 exp 2( )( ) 1
,( )2 sin( )( ) exp 2 exp 2
i ig xg x
g xig x i i
115
but the first and second rows of (13.51) are (i) and (ii) respectively so the proof is done.
This theorem gives explicit connection relations between the solutions ( ),g x ( ),g x
(1) ( ),g x and (2) ( )g x to the ( )g x equation (13.1). In the next section, we’ll show how
properties of the ( )g x equation (13.1) and its solutions can be used to obtain connection
coefficients appearing in linear dependence relations between our solutions
1 2 3( ), ( ), ( ),y x y x y x and 4 ( )y x to our 41 x potential SL equation (8.5).
116
Section 14: The Connection Problem
In this section we’ll make suitable changes of variables, parameter restrictions, and
contour choices to relate solutions to the g equation (13.1) to solutions of our 41 x
potential SL equation (8.5). This work will enable the connection formulas obtained in
Theorem 8 for the g equation (13.1) to be reformulated so that they yield corresponding
connection formulas for our 41 x potential SL equation (8.5). First we need to make the
following choices for the parameters , , and k in (13.1):
(14.1) 0, ,i k i
As the connection relations in Theorem 10 are invariant under changes to Bühring’s
parameter introduced to ease his numerical calculations, we elect here to set this
parameter to zero and point out that this parameter is not to be confused with our eigen
parameter from equation (8.5) which at present remains arbitrary in . Also, as
Bühring’s x variable from Section 13 is different from our own x variable from
Sections 8-12, to avoid confusion, we elect here to change his x variable to .s Under
these choices, the equations (13.1)-(13.2), (13.4)-(13.6), (13.12)-(13.13), (13.15)-(13.17),
(13.19)-(13.21), (13.23), (13.34)-(13.39) become
(14.2) 2 2 21
( ) ( ) ( ) 0,4
s g s sg s s s g s
(14.3) 2
2( ) , , \ 0 ,n
n
n
n
g s d s s
117
(14.4) 2 2 ,n
n nd d
(14.5) exp ( ),2
i ig g s
s
(valid also with ),
or equivalently, ( ),i
g i g ss
(valid also with ),
(14.6) 1
( ) exp( ) ( ) ,2
C
g s st v t dti
(14.7) 2
4 3 4( ) ( ) ( ) 0,4
ss g s s g s s g s
(14.8) 2
4 3 4( ) ( ) ( ) 0,4
ss W s s W s s W s
(14.9)
4 3 22
4 3 2
2 (4) (3)
( ) 1 ( ) ( ) 1 4 ( ) ( )
1 ( ) 7 35 4 ( ) ( ) 0,
t
d d dM v t t v t t v t v t v t
dt dt dt
t v t tv v t v t
(14.10)
2 3 2 2
2
( 1) 2 2 3 2 3
( , ( )) exp( ) .12 ( )
4
st
t s v s v sv v t s v sv v sv v
B e v t stts v tsv tv sv v sv v
(14.11) 1 2
2( ) ,n
n
n
n
v t t b t
(14.12) 2 2
( 1 2 ), ,
( 1)m m
mb d
(14.13)
3
1( ) ( 1) exp( ) ( ) , arg ,
2 22C
g s st v t dt si
118
(14.14) 1 2
0
( ) 1 1 2 1 1 ,n
n
n
v t ti A ti F ti
(14.15) 1 2
0
( ) 1 1 2 1 1 ,n
n
n
v t ti A ti F ti
(14.16)
exp ( 2)( 1) ( ) csc( ) ( ) , 2
( ) exp ( 2)( 1) ( ) csc( ) ( ) , 2
( 1) ( ), 1
i iv t u t t i
w t i iv t u t t i
v t t
,
(14.17)
(1 )/21
( ) sin( ) ( 1) ( 1): ,
2 ( 4) (2 )
i F G
v i
(14.18)
1
(1) 1( ) exp( ) ( ) , 2 arg 2,
2C
g s st w t dt si
(14.19)
2
(2) 1( ) exp( ) ( ) , 2 arg 2,
2C
g s st w t dt si
(14.20) ( ) : ( ), 2,w t v t t i
and
(14.21) ( ) : ( ), 2.w t v t t i
Note: In (14.5), we have used exp( 2)( ) ( ) .i i
Relations (13.40) and (13.18) are
(14.22) 1 2
2( ) ( 1) ( ) ( 1) ,n
n
n
n
v t v t t b t
, 1.t
Here are roots of the Hill determinant arising from the three-term recurrence relation
119
(14.23) 2
2 2 2 2 2
12 0
4n n nn d d d
, , 2, 1,0,1,2,n .
Theorem 10 on connection formulas for (1) (2)( ), ( ), ( ),g s g s g s and ( )g s becomes
Theorem 11: In the sector 2 arg 2,s the solutions to equation (14.2), ( ),g s
( ),g s (1) ( ),g s and
(2) ( ),g s satisfy the linear connection relations,
(i) (1) (2)( ) exp 2 ( ) exp 2 ( ), ,g s i g s i g s
(ii) (1) 1( ) exp ( 2) ( ) exp ( 2) ( ) ,
2 sing s i g s i g s
i v
and
(iii) (2) 1( ) exp ( 2) ( ) exp ( 2) ( ) .
2 sing s i g s i g s
i v
This theorem gives explicit connection relations between the solutions ( ),g s ( ),g s
(1) ( ),g s and (2) ( )g s to the ( )g s equation (14.2). The proof of Theorem 11 follows
immediately from Theorem 10 after taking the notation choices of this section, the
parameter restrictions in (14.1), and Bühring’s numerical parameter 0 . The
restriction 2 arg( ) 2s above in Theorem 11, (i)-(iii), is imposed to meet the
contour requirement from Theorem 9. That is, when 2 arg( ) 2,s on account of
a decaying exponential, the bilinear concomitant expression (14.10) is driven to 0 at the
ends of each of the contours 1 2, ,C C and 3C and hence Ince’s requirement of contour is
satisfied. As we now intend to use properties of solutions to equation (14.2) to formulate
analogous properties that will hold for equation (8.5), we first must establish a connection
120
between these two equations. To this end, from equation (14.2), observe that any of the
four changes of variables
(14.24) 1
2 ,s x ( ) ( )g s G x OR ,s i x ( ) ( )g s G x
takes equation (14.2) into the differential equation
(14.25) 2 2
2
1 1( ) ( ) ( ) 0.
4x G x xG x x G x
x
From this observation if follows that for any ( )g s satisfying equation (14.2), all of
( ), ( ), ( ),g i x g i x g x and ( )g x satisfy equation (14.25) provided that these
functions are defined. Next we establish that solutions of equation (14.25) in turn give
rise to solutions to equation (8.5). From equation (14.25), the transformation
(14.26) 1
2( ) ( )G x x y x
yields equation (8.5). This means that for any ( )G x satisfying equation (14.25),
12( ) ( )y x x G x satisfies equation (8.5). In equations (14.24)-(14.26), the x variable
used is now the same x variable that appears in Sections 8-12, 0, .x Now that the
connection between the equations (14.2), (14.25), and (8.5) has been made. We
summarize in the figure on the next page.
121
Figure 2Related Equations, Transformations, and Parameter Restrictions
Bühring’s ( )g s ODE:
(13.1): 2
2 2 2 2 21( ) ( ) ( ) 02
s g s sg s s k s g s
, 0 ,k i i
(14.2): 2 2
2
1( ) ( ) ( ) 0
4s g s sg s s g s
s
, ( ) ( )s i x g s G x OR , ( ) ( )s x g s G x
(14.25): 2 2
2
1 1( ) ( ) ( ) 0
4x G x xG x x G x
x
12( ) ( )G x x y x
41 x Potential SL Equation:
(8.5): 4
1( ) ( ) ( )y x y x y x
x
122
Figure 2 clarifies that by use of the parameter restrictions (14.1) coupled with the changes
of variables (14.24) and then (14.26), solutions to equation (13.1) can be used to define
new functions which will satisfy equation (8.5). Since the choice of contour in Theorem
9 is quite flexible and bound only by the requirement that the bilinear concomitant is the
same at both ends, for reasons associated with the two separate substitutions for s that
we intend to make, s i x and ,s x we introduce for our purposes the following
new contours.
Figure 3: Suitable Contours II
6 4 5C C C
4C 6C 4C
5C 5C
These new contours initiate and terminate in the 1st quadrant of the t plane with the
linear pieces inclined at an angle 30 from the positive real t axis though any
positive angle for which 0 45 would be sufficient for our needs. We’ll now
show that these new contours satisfy the bilinear concomitant requirement from Theorem
9 after either of the the changes of variables s i x or .s x
123
Lemma 5: Let the bilinear concomitant expression for any generalized Laplace integral
solution to the ( )g s equation (14.2) be given as in (14.10).
(i) After the substitution s i x in (14.10), the bilinear concomitant expression tends to
0 at the ends of all of the contours 4 5 6, , ,C C C and 6C provided that (0, ).x
(ii) After the substitution s x in (14.10), the bilinear concomitant expression tends
to 0 at the ends of all of the contours 4 5 6, , ,C C C and 6C provided that (0, )x and
0 arg 2.
PROOF: Let (0, ).x We substitute s i x into (14.10) obtaining the concomitant
(14.27)
2
3 2 2
2
21 2 3 2 3
exp .1
24
iv v iv v iv ivt v t v v
x x x x x xit
x tv itv iv ivtv v v
x x x x
Using the asymptotic behavior for the Frobenius solutions ( )v t as t in [23, p.
1456] for t belonging to any of 4 5 6, , ,C C C or 6C , in the limit as ,t any
derivative of v is dominated by a rational function of t multiplied by 1 2exp K t for
some postitive contant .K Therefore, to show (14.27) tends to 0 at the ends of the
contours 4 5 6, , ,C C C and 6 ,C it will be sufficient to show that the exponential factor
exp it x which dominates all such terms is a decaying exponential. We let 1 2t t it
so that 1 2exp exp exp .it x it x t x The first factor is bounded by 1 whereas the
124
second factor 2exp 0t x as both 2 ,t t as is case at the ends of all of the
contours 4 5 6, , ,C C C and 6C so (i) is established. Now let (0, )x and
0 arg( ) 2. We substitute s x into (14.10) obtaining the concomitant
(14.28)
32 3 2 22
2
1 2 2 3 2 3
exp .1
24
t x v x v xv v t x v xv v xv v
xt
t x v t xv tv xv v xv v
As before, to show that this expression tends to 0 at the ends of the contours
4 5 6, , ,C C C and 6 ,C it is sufficient to show that the exponential factor exp xt is a
decaying exponential. We let ire and 1 2t t it and so that
(14.29)
12
1 2
1 12 2
1 2 1 2
exp exp cos 2 sin 2
exp cos 2 sin 2 exp sin 2 cos 2 .
xt xr i t it
xr t t ixr t t
The second factor above is bounded by 1 whereas the first factor is a decaying
exponential if 1 2cos 2 sin 2 0t t and 1 2cos 2 sin 2t t at the
ends of the contours. To see that this is the case : 0 arg 2, we solve for
2t in the inequality above obtaining 2 1cot 2 .t t Observe that for 0 2, this
inequality is satisfied in the sector of the t plane, 2 arg 4,t in which all of
the contours 4 5 6, , ,C C C and 6C intiate and terminate. Now we must show that
125
1 2cos 2 sin 2t t at the ends of these contours. At the ends of these
contours, 2 sin ,t t
1 cos ,t t so 1 2cos 2 sin 2t t may be approximated
for large t by cos 2 cos sin 2 sin cos 2t t as t
: 0 arg 2. This analysis validates that indeed, after the substitution
,s x the concomitant (14.10) tends to 0 at the ends of all of the contours
4 5 6, , ,C C C and 6C so (ii) is established and the proof of this lemma is done.
Lemma 5 establishes that these new contours in Figure 3 are indeed suitable for defining
Laplace integral solutions to (14.2) that will be valid after either of the substitutions
s i x and ,s x when 0, , 0 arg 2.x We point out here that
Bühring’s original contours 1 2, ,C C and 3C were insufficient for our needs since after
either of the substitutions s i x or s x is made into the bilinear concomitant
expression (14.10), this expression does not tend to 0 at the ends of contours 1 2, ,C C
and 3C for when (0, ), 0 arg 2x , and hence the requirement on contour from
Ince’s theory would not be met.
We define the new Laplace integral solutions to equation (14.2) as
(14.30) 4
(3) 1( ) : exp ( ) ,
2C
g s st w t dti
3 arg( ) 4 3,s
and
126
(14.31) 5
(4) 1( ) : exp ( ) ,
2C
g s st w t dti
3 arg( ) 4 3.s
By rotation of contour 3 ,C the Laplace integral for for ( )g s valid in the sector
3 arg( ) 4 3s is
(14.32)
6
1( ) ( 1) exp( ) ( ) , 3 arg( ) 4 3, .
2C
g s st v t dt si
We now state as a theorem the linear connection relations that hold for (3) ( ),g s
(4) ( ),g s
( ),g s and ( ),g s equations (14.30-14.32).
Theorem 12: In the sector 3 arg( ) 4 3,s the solutions to equation (14.2),
( ),g s ( ),g s (3) ( ),g s and
(4) ( ),g s satisfy the linear connection relations,
(i) (3) (4)( ) exp 2 ( ) exp 2 ( ), ,g s i g s i g s
(ii) (3) 1( ) exp ( 2) ( ) exp ( 2) ( ) ,
2 sing s i g s i g s
i v
and
(iii) (4) 1( ) exp ( 2) ( ) exp ( 2) ( ) .
2 sing s i g s i g s
i v
PROOF: From the generalized Laplace integral (14.6), we insert ( )w t in for ( )v t and
choose contour 6.C We compare with (14.32) and since 1t 6 ,t C we have
( ) ( 1) ( )w t v t as given in relation (14.16) so the LHS is ( ).g s Now on the
RHS, we apply a deformation of the contour argument and replace contour 6C with 6C
127
which gives the relation
(14.33)
6
4
5
1( ) ( 1) exp( ) ( )
2
1exp( ) ( )
2
1exp( ) ( )
2
1exp( ) ( ) .
2
C
C
C
g s st v t dti
st w t dti
st w t dti
st w t dti
Let r be the distance from the origin to the midpoint of the line segment , let the
length of be 2 , and pass r to . In this limit, only the 1st and 3rd terms above
contribute to integral. To see this, we parametrize 1 2: ( ) ( ) ( )t t it to find
(14.34)
1 2( ) ( )
( ) 3 2 2 2 3 2 3 , 0 1.
t t
t r i r
Since 1t as ,r we employ (14.16) in the integral over and then consider
(14.35) 1
1 2
0
lim exp( ) ( ) lim exp( )exp( ) ( ( )) ( ).r r
st v t dt st ist v t dt
Bounding the integral on the RHS we find that
(14.36)
1 1
1 2 1 2
0 0
1
1 2
0
exp( )exp( ) ( ( )) ( ) exp( ) exp( ) ( ( )) ( )
1 3 exp(Re[ ] Im[ ] ) ( ( )) .
st ist v t dt st ist v t dt
i s t s t v t d
For large ,t we apply the approximations 2 sin 2,t t t
1 cos 3 2,t t t
128
so for ,r we have
1 2
3 Re Im[ ]Re Im[ ] .
2
s ss t s t t
Since
Im[ ] 3 Re[ ]s s s in the sector 3 arg( ) 4 3,s for each fixed s in this sector,
1 2exp Re Im[ ] exp ,s t s t O K t t for some positive scalar K which
depends on the fixed choice of .s Finally, we use the rate of growth bound in [23, p.
1456] (compare proof of Theorem 10) for v as t , 1 2
exp( ( )) ,
( )
M tv t
t
,t for some positive constant .M Thus for each fixed s in the sector
3 arg( ) 4 3,s as ,r the decaying exponential 1 2exp Re Im[ ]s t s t
dominates ( ( ))v t and drives the contour integral over to 0 so
(14.37) 1
lim exp( ) ( ) 0.2r
st w t dti
From relations (14.30)-(14.31), as ,r we will now show the RHS of (14.33) is a
linear combination of (3) ( )g s and
(4) ( ).g s Observe from (14.30)-(14.31), (14.20)-
(14.21), and (14.37),
(14.38)
6
4 5
4 5
1( ) exp( ) ( )
2
1 1lim exp( ) ( ) exp( ) ( )
2 2
1 1exp( ) ( ) exp( ) ( ) .
2 2
C
rC C
C C
g s st w t dti
st w t dt st w t dti i
st w t dt st w t dti i
129
Next we deform contours 4C and 5C so that they approach and depart from the
singularities that they enclose along parallel coinciding rays that are inclined 30 from
the positive 1t axis and feature counter-clockwise radius circles, 4t i
C
and 5 ,t i
C
around the singularities t i and t i respectively. The contour integration along the
oppositely-oriented coinciding rays cancel. For 2, we make use of (14.16) to obtain
(14.39)
4 5
4 5
4
5
1 1exp( ) ( ) exp( ) ( )
2 2
1 1exp( ) ( ) exp( ) ( )
2 2
1 1exp( ) exp ( 2)( 1) exp 2 ( ) ( )
2 sin( ( ))
1exp( )
2
t i t i
t i
t i
C C
C C
C
C
st w t dt st w t dti i
st w t dt st w t dti i
st i i v t u t dti
sti
1
exp ( 2)( 1) exp 2 ( ) ( ) .sin( ( ))
i i v t u t dt
Again we find that two of the above contour integrals are zero by the Cauchy-Goursat
Theorem; namely, the contour integrals with integrands containing ( )u t and ( )u t
as
the integrands for these contour integrals are analytic in the vicinity of the singularities
t i and t i respectively. The contour integrals containing ( )v t and ( )v t
remain
so the RHS is
130
(14.40)
4 5
4
5
1 1exp( ) ( ) exp( ) ( )
2 2
1exp exp( ) ( )
2 2
1exp exp( ) ( ) .
2 2
C C
C
C
st w t dt st w t dti i
ist v t dt
i
ist v t dt
i
Lastly we invoke relations (14.20)-(14.21) to obtain
(14.41)
4 5
4 5
(3) (4)
1 1exp( ) ( ) exp( ) ( )
2 2
1 1exp exp( ) ( ) exp exp( ) ( )
2 2 2 2
exp ( ) exp ( ),2 2
C C
C C
st w t dt st w t dti i
i ist w t dt st w t dt
i i
i ig s g s
so (i) is established. To see that (i) (ii) and (iii), we write relation (i) for in
the form
(14.42)
(3)
(4)
1 exp 2 1 exp 2( ) ( )
( ) ( )1 exp 2 1 exp 2
i ig s g s
g s g si i
.
Assuming that the characteristic index is not an integer, the coefficient matrix above is
nonsingular and we can solve the system to obtain
(14.43)
(3)
(4)
exp 2 exp 2( )( ) 1
,( )2 sin( )( ) exp 2 exp 2
i ig sg s
g sig s i i
but the first and second rows of (14.43) are (i) and (ii) respectively so the proof is done.
131
Next we show that asymptotics of the solutions (3) ( )g s and
(4) ( )g s defined in (14.30)
and (14.31) as s in the sector 3 arg( ) 4 3s can be obtained by applying a
well-known Lemma of Barnes [22] that is also commonly referred to in the literature as
“Watson’s Lemma for Loop Integrals”. We give its formulation in [105, p. 48-49] below.
Barnes’ Lemma (on a rotated contour): Consider the loop integral
(14.44) 1
( ) ( ) ,2
st
C
iI s t f t e dt
where C is the contour as shown which loops the origin in the positive sense and
features oppositely-directed parallel rays, to and from the origin, inclined at an angle of
degrees from the positive real t axis such that Re[ ] 0st towards both termini of .C
Assume all of the following: t plane
(i) ( )f t is continuous on and analytic within contour .C
(ii) The integral (14.44) is convergent. C
(iii) 0
( ) ( )n
n
n
f t c t
is convergent for , 0.t inclination angle
In the sector of the s plane, 2 arg( ) 2 ,ise 0, the loop integral
(14.44) has uniformly in arg( )s the asymptotic series representation,
(14.45) 0
( ) ~ , .(1 )
nnn
n
cI s s s
n
The proof of this lemma was first given by E.W. Barnes in [16, p.254-256].
132
By use of this lemma, we will now derive aymptotics for (3) ( )g s and
(4) ( )g s as defined
in (14.30)-(14.31), the result of which, is given as a theorem.
Theorem 13: In the sector 3 arg( ) 4 3 ,s 0,
(14.46) (3)
1 20
( 1)( ) ~ exp( ) , ,
( )
n
n
nn
ag s i is s
is
and
(14.47) (4)
1 20
( 1)( ) ~ exp( ) , ,
( )
n
n
nn
ag s i is s
is
uniformly for 3 arg( ) 4 3 ,s 0.
PROOF: In the definitions for (3) ( )g s and
(4) ( ),g s equations (14.30)-(14.31), we use the
relations (14.20)-(14.21), (14.14)-(14.15) to obtain for 3 arg( ) 4 3,s
(14.48) 4
1(3) 2
0
1( ) 1 (1 ) exp( ) , 2,
2 (1 2 )
nn
nC
ag s ti it st dt t i
i n
and
(14.49) 5
1(4) 2
0
1( ) 1 (1 ) exp( ) , 2,
2 (1 2 )
nn
nC
ag s ti it st dt t i
i n
where we have used the relation ( 1/ 2) (1 2 ).n nA a n We now conform
(14.48)-(14.49) to the formulation of Barnes’ Lemma given above. Thus we change the
variable of integration in (14.48) using the substitution 1 it t which acts to translate
contour 4C to the origin and then rotates 90 clockwise yielding contour 4C in the t
133
plane ( 60 ). Similarly, we change the variable of integration in (14.49) using the
substitution 1 it t which acts to translate contour 5C to the origin and then rotates
90 counterclockwise yielding contour 5C in the t plane ( 120 ). After these
changes of variables, (14.48)-(14.49) for 3 arg( ) 4 3s become
(14.50) 4
1(3) 2
0
1( ) exp ( ) exp( ) , 2,
2
n
n
nC
g s is t c t ist dt t
and
(14.51) 5
1(4) 2
0
1( ) exp( ) ( ) exp( ) , 2,
2
n
n
nC
g s is t c t ist dt t
where (1 2 ).n nc a n We let is s in (14.50) and let is s in (14.51) to obtain
(14.52)
4
1(3) 2
0
1 5 11exp ( ) exp( ) , 2, arg ,
2 6 6
n
n
nC
sg s t c t s t dt t s
i
and
(14.53)
5
1(4) 2
0
1 5exp( ) ( ) exp( ) , 2, arg .
2 6 6
n
n
nC
sg s t c t s t dt t s
i
Now we let s s in (14.52) and let s s in (14.53) to obtain
134
(14.54)
4
1
1(3) 2
0
( )
5exp ( ) exp( ) , 2, arg ,
2 6 6
n
n
nC
J s
s ig s t c t s t dt t s
i i
and
(14.55)
5
2
1(4) 2
0
( )
5 11exp( ) ( ) exp( ) , 2, arg .
2 6 6
n
n
nC
J s
s ig s t c t s t dt t s
i i
The functions 1( )J s and 2 ( )J s meet the assumptions of Barnes’ Lemma so by
comparison to (14.44), we identify that 1 2. We apply the lemma to obtain
(14.56) 1
(3) 2
0
( ) ~ exp( ) ( ) , ,(1 2 ) (1 2 )
nn
n
ag s is is s
i n n
and
(14.57) 1
(4) 2
0
( ) ~ exp( ) ( ) , ,(1 2 ) (1 2 )
nn
n
ag s is is s
i n n
uniformly for 3 arg( ) 4 3,s 0.
Lastly we employ in (14.56)-(14.57) the identity
(14.58) (1 2 ) (1 2 ) ( 1) ,nn n
and simplify to obtain the relations (14.46)-(14.47) and the proof is done.
135
Remark 8: We find that the asymptotics (14.46)-(14.47) obtained in Theorem 13 are in
agreement with those given by Bühring for his functions (1) ( )g x and
(2) ( )g x [23, p.
1456, eq. 81a, 81b, 83], after the parameter restrictions (14.1), 0, and the variable
choice x s has been made in (13.36)-(13.37) though, due to Bühring’s original contour
choices 1,C 2 ,C his aysmptotics (like his Laplace integrals), are valid in the different
sector of the s plane, 2 arg( ) 2s (He does assert that extended domains of
validity for analytically continued (1) ( )g x and
(2) ( )g x functions are possible by means of
a contour rotation argument [23, p. 1456] but explicit formulas for these continued
functions are not given.).
Next we define the following solutions to equation (14.28) for when (0, ) :x
(14.59)
4
(1) (3) 1( , ) : ( ) exp( ) ( ) , ,
2C
G x g i x it x w t dti
(14.60)
5
(2) (4) 1( , ) : ( ) exp( ) ( ) , ,
2C
G x g i x it x w t dti
(14.61)
5
(3) (4) 1( , ) : ( ) exp( ) ( ) , : 0 arg( ) 2,
2C
G x g x xt w t dti
and
(14.62)
4
(4) (3) 1( , ) : ( ) exp( ) ( ) , : 0 arg( ) 2.
2C
G x g x xt w t dti
We now introduce two suitably-normalized solutions to (8.5) defined about 0x as
136
(14.63) 1 1
(1) (1) (3)2 2( , ) : ( , ) , 0, , ,i
y x ix G x ix g xx
(14.64) 1 1
(2) (2) (4)2 2( , ) : ( , ) , 0, , .i
y x x G x x g xx
For (0, ), : 0 arg( ) 2x , we introduce two suitably-normalized solutions
to (8.5) defined near x as
(14.65) 1 1
(3) (3) (4)2 2( , ) : ( ) ( , ) ( ) ( ),y x i i x G x i i x g x
and
(14.66) 1 1
(4) (4) (3)2 2( , ) : ( ) ( , ) ( ) ( ).y x i i x G x i i x g x
We will now establish a relationship between the two sets of solutions,
( ) ( , ), 1,2,3,4iy x i and ( , ), 1,2,3,4.iy x i As a motivator for this, we begin by
proving the following lemma regarding asymptotic relations for the these two sets of
solutions to equation (8.5) where it is understood that (0, ).x
137
Lemma 6: Regarding the solutions to (8.5), ( ) ( , )iy x and ( , ),iy x for 1,2,3,4 :i
(i) (1) ( , )y x and 1( , ),y x defined by the relations (14.63) and (8.97), Theorem 4 in
Section 8, have identical asymptotic series representations near 0, .x
(ii) (2) ( , )y x and 2 ( , ),y x defined by the relations (14.64) and (8.111), have identical
asymptotic series representations near 0, .x
(iii) (3) ( , )y x and 3( , ),y x defined by the relations (14.65) and (10.28), have identical
asymptotic series representations near , : 0 arg( ) 2.x
(iv) (4) ( , )y x and 4 ( , ),y x defined by the relations (14.66) and (10.39), have identical
asymptotic series representations near , : 0 arg( ) 2.x
PROOF: In the asymptotic relations (14.46)-(14.47), we put s i x and simplify to
obtain for (0, )x and ,
(14.67) 1
(3) 2
0
( ) ~ exp( 1 ) ( 1) , 0 ,n n
n
n
g i x ix x a x x
and
(14.68) 1
(4) 2
0
( ) ~ exp(1 ) , 0 .n
n
n
g i x x x a x x
Similarly, in the asymptotic relations (14.46)-(14.47), we put s x and simplify to
obtain for (0, )x and for : 0 arg( ) 2,
138
(14.69) 1
(3) 2
0
( ) ~ ( ) exp( ) ( 1) ( ) , ,n n
n
n
g x i i x i x a i x x
(14.70) 1
2(4)
0
( ) ~ exp( ) ( 1) ( ) , .n n
n
n
g x i i x i x a i x x
Now from relations (14.63)-(14.66) and (14.67)-(14.70), we obtain for (0, ),x
(14.71) (1)
0
( , ) ~ exp( 1 ) ( 1) , 0 ,n n
n
n
y x x x a x x
(14.72) (2)
0
( , ) ~ exp(1 ) , 0 , ,n
n
n
y x x x a x x
and for (0, )x and : 0 arg( ) 2,
(14.73) (3)
0
( , ) ~ exp( ) ( 1) ( ) , ,n n
n
n
y x i x a i x x
(14.74) (4)
0
( , ) ~ exp( ) ( 1) ( ) , .n n
n
n
y x i x a i x x
The recurrence relation for na is given in [23, p. 1456, eq. 83]. With the parameter
restrictions (14.1) and after setting Bühring’s parameter to zero, this relation is
(14.75) 1 3 0 1 2
1, 1, 0, 1,2,3,
2 2n n n
na a a a a a n
n
.
We use (14.75) in (14.71)-(14.74) obtaining for (0, ), ,x 0 ,x
(14.76) 3 4 5 2
(1) 6 75( , ) ~ exp( 1 ) 1 ( ) ,
6 4 2 4 72
x x xy x x x x O x
(14.77) 3 4 5 2
(2) 6 75( , ) ~ exp(1 ) 1 ( ) ,
6 4 2 4 72
x x xy x x x x O x
139
and for (0, ),x : 0 arg( ) 2, as ,x
(14.78)
(3) 3 4 5 6 7
3 22
1 1 5( , ) ~ exp( ) 1 ( ) ,
4 72 46 2
i iy x i x x x x x O x
(14.79)
(4) 3 4 5 6 7
3 22
1 1 5( , ) ~ exp( ) 1 ( ) ,
4 72 46 2
i iy x i x x x x x O x
Lastly, comparison of (14.76)-(14.79) with the asymptotic relations for
( , ), 1,2,3,4,iy x i given by the relations (11.25), (12.19)-(12.21) finishes the proof.
Now that it has been established that an asymptotic relation for each of the solutions
( ) ( , ), 1,2,3,4iy x i matches to an asymptotic relation for one of the solutions
( , ), 1,2,3,4,iy x i we may expect that some of these solutions could be identical.
We will now prove a theorem confirming that this is indeed the case; namely, we show
( , ) (0, ) ,x (1)
1( , ) ( , )y x y x and (0, ), : 0 arg 2,x
(3)
3( , ) ( , ).y x y x
140
Theorem 14: Let the solutions (1) ( , ),y x and
(3) ( , )y x
be given as in equations (12.19)-(12.21), (11.25), (14.76), and (14.78). Then
(i) (1)
1( , ) ( , ),y x y x ( , ) (0, ) .x
(ii) (3)
3( , ) ( , ),y x y x (0, ), : 0 arg 2.x
PROOF: Since (1) ( , )y x is a solution to (8.1) defined about 0,x
(1) ( , )y x must be a
linear combination of the two solutions from our fundamental system that we derived
near this endpoint consisting of the solutions given in (8.97), Theorem 4 in Section 8, and
(8.111). That is for ( , ) (0, ) ,x
(14.80) (1)
1 2
3 3 3
( , ) ( , )( , )
exp( 1 ) 1 ( ) exp( 1 ) 1 ( ) exp 1 1 ( )
y x y xy x
x x O x A x x O x B x x O x
where A and B are scalars. To see that 0B we rely on subdominance. If 0,B then
the asymptotic for (1) ( , )y x near 0x is B times the asymptotic for . This is
not the case so 0.B To see that 1A we assume 1A but this would imply that the
asymptotic for (1) ( , )y x near is A times the asymptotic for but again this
is not the case as (1) ( , )y x and have identical asymptotics. We have thus
shown that (1)
1( , ) ( , )y x y x ( , ) (0, )x so (i) has been established. We note
here that the reason that these two solutions correlate identically is no coincidence but
rather due to the normalization that we chose for the constant E appearing back in
Lemma 4 as well as in Theorem 6. This normalization was chosen to produce a solution
( , ), 1,2,3,4,iy x i
2 ( , )y x
0x 1( , )y x
1( , )y x
141
whose asymptotic conformed to that given by the standard F.P.S. ansatz. To show that
(3)
3( , ) ( , ),y x y x (0, ), : 0 arg 2,x we employ a similar argument.
Since (3) ( , )y x is a solution to (8.1) defined near ,x
(3) ( , )y x must be a linear
combination of the two solutions from our fundamental system that we derived near this
endpoint consisting of the solutions in (10.28) and (10.39). That is
(14.81) (3)
3 4
3 3 3
( , ) ( , )( , )
exp( ) 1 ( ) exp( ) 1 ( ) exp 1 ( )
y x y xy x
i x O x C i x O x D i x O x
where C and D are scalars. To see that 0,D we again rely on subdominance. If
0,D then the asymptotic for (3) ( , )y x near x is D times the asymptotic for
4 ( , ).y x This is not the case so 0.D To see that 1C we assume 1C but this
would imply that the asymptotic for (3) ( , )y x near x is C times the asymptotic for
3( , )y x but again this is not the case as (3) ( , )y x and 3( , )y x have identical
asymptotics. Therefore (3)
3( , ) ( , )y x y x (0, ), : 0 arg 2,x so (ii)
has been established and the theorem is now proven.
We point out here that a similar such subdominance argument does not establish equality
between either of the dominant pairs of solutions 2 ( , )y x and (2) ( , )y x or 4 ( , )y x and
(4) ( , ).y x However, we can prove from the matching asymptotics of 2 ( , )y x and
(2) ( , )y x near 0x as well as those of 4 ( , )y x and (4) ( , )y x near x the
following theorem of linear dependence.
142
Theorem 15: Let the solutions (2) ( , ),y x and
(4) ( , )y x
be given as in equations (12.19)-(12.21), (11.25), (14.77), and (14.79). Then
(i) (2)
2 1( , ) 1 ( , ) ( ) ( , ), ( , ) (0, ) .y x y x y x x
(ii) (4)
4 3( , ) 1 ( , ) ( ) ( , ), (0, ), : 0 arg( ) 2.y x y x y x x
Here ( ) and ( ) are unknown constants.
PROOF: Since (1) (2)( , ), ( , )y x y x is a fundamental system to (8.5) near 0,x
( , ) (0, ) ,x our 2 ( , )y x solution defined near 0x must be a linear
combination of these solutions or ( , ) (0, ) .x
(14.82) (2)
2 1( , ) ( ) ( , ) ( ) ( , ),y x y x y x
where ( ) and ( ) are constants. The determination that 1 comes from the fact
that 2 ( , )y x and (2) ( , )y x have identical asymptotic relations near 0x as has been
shown in Lemma 6. Observe that if 1, then (14.82) contradicts the fact that the
asymptotics near 0x for 2 ( , )y x and (2) ( , )y x match identically so 1 . We have
thus established (i). Unfortunately, ( ) in relation (14.82) cannot be directly
determined by use of asymptotics near 0x as the term 1
1( ) ( , ) ( ),xy x O xe
0 ,x and hence it tends rapidly to zero as 0 .x To establish (ii), we utilize a
similar argument. Since (3) (4)( , ), ( , )y x y x is a fundamental system to (8.5) near
,x for (0, ),x : 0 arg( ) 2, our 4 ( , )y x solution defined near x
must be a linear
( , ), 1,2,3,4,iy x i
143
combination of these solutions. That is for (0, ), : 0 arg( ) 2,x
(14.83) (4)
4 3( , ) ( ) ( , ) ( ) ( , ),y x y x y x
where ( ) and ( ) are constants. The determination that 1 comes from the fact
that 4 ( , )y x and (4) ( , )y x have identical asymptotic relations near x as has been
shown in Lemma 6. Observe that if 1 , then (14.83) contradicts the fact that the
asymptotics near x for 4 ( , )y x and (4) ( , )y x match identically so 1. We
similarly find that ( ) in relation (14.83) cannot be directly determined by use of
asymptotics near x as the term 3( ) ( , ) (exp( ))y x O i x and hence it tends
rapidly to zero as ,x with Im 0. As (ii) has now been established, the
theorem is proven.
Next we make use of the results proven in Theorems 14 and 15 to obtain the asymptotic
relation near x for our solution 1( , )y x as well as the asymptotic relation near
0x for our solution 3( , ).y x From relations (14.63), (14.65), Theorem 14, and
Theorem 15, it is evident that these asymptotic relations can be obtained directly from the
asymptotic relation near x for (3)g i x and from the asymptotic relation near 0x
for (4) ( ).g x We clarify here that equation (14.84) is valid for ( , ) (0, )x ,
whereas equations (14.85)-(14.89) are valid for (0, ), : 0 arg( ) 2.x
We start with the linear relations (ii)-(iii), in Theorem 12, and apply them for the
functions (3)g i x and (4) ( )g x giving
144
(14.84) (3) 1( ) exp 2 ( ) exp 2 ( ) ,
2 sing i x i g i x i g i x
i
and
(14.85)
(4) 1( ) exp 2 ( ) exp 2 ( ) ,
2 sing x i g x i g x
i
Next we use the symmetry relation (14.5) to eliminate the functions ( ),g i x ( ),g i x
12( ),g x and
12( )g x that appear in relations (14.84)-(14.85). This gives
(14.86) (3) 1 2 1 21( ) ( ) ( ) ,
2 sing i x g x g x
i
and
(14.87)
(4) 1 2 1 21( ) exp[ ] ( ) exp[ ] ( ) ,
2 sing x i g i x i g i x
i
Next we use (i) in Theorem 12 to eliminate the functions ,g x ,g x
( ),g i x and ( )g i x that appear in relations (14.86)-(14.87). This gives for all
(0, )x and all : 0 arg( ) 2C
(14.88) (3) (3) (4)
1 2( ) ( , ) ( ) ( , ) ( ),g i x R g x R g x
145
(14.89) (4) (3) (4)
3 1( ) ( , ) ( ) ( , ) ( ),g x R g i x R g i x
where we find the dependent constants , , 1,2,3iR i to be
(14.90)
1 12 2
1
1( , ) exp( 2) exp( 2)
2 sin( )
1,
2 sin( )
R i ii
ii
i
1 12 2
2
1( , ) exp( 2) exp( 2)
2 sin( )
1,
2 sin( )
R i ii
ii
i
1 12 2
3
1( , ) exp( 3 2) exp(3 2)
2 sin( )
1.
2 sin( )
R i ii
ii
i
Interestingly, the constants ( , ), 1,2,3,iR i satisfy the relation 2
1 2 3 1.R R R
Remark 9: We find the contants ( , ), 1,2,3,R i to be related to the connection
coefficients 13 14, ,D D 23D , and 24D , that Bühring obtained in [23, p. 1457, eq. 95a-95d]
for his equation studied after the parameter restrictions (14.1) and with his parameter
set equal to 0. Particularly, after these restrictions are made, we find that
(14.91) 1
2
1 1 13( , ) ,R R i D
1
21
2 2 14( , ) ,R R i i D
1
2
3 3 23( , ) ,R R i D
or 1 1
2 213 1 14 2( ) , ( ) ,D i R D i i R
146
1
223 3 24 13( ) , ( ).D i R D D
and that Bühring’s connection coefficients satisfy the relation 1
2 213 14 23D D D i .
Using the asymptotic relations (14.67)-(14.70) in (14.88)-(14.89), we find the
asymptotics for (3) ( )g i x near x and (4) ( )g x near 0x as
(14.92) 1
(3) 21
0
( ) ~ ( ) exp( ) ( 1) ( ) ,n n
n
n
g i x iR i x i x a i x x
and
(14.93) 1
(4) 21
0
( ) ~ exp(1 ) , 0n
n
n
g x R x x a x x
Finally, by relations (14.63), (14.65) & (14.92)-(14.93) we obtain the two asymptotics
(14.94)
1
1(1) 2
1
0( )
( , ) ~ ( ) exp( ) ( 1) ( ) ,n n
n
nK
y x R i i x a i x x
,
and
(14.95)
2
1(3) 2
1
0( )
( , ) ~ ( ) exp(1 ) , 0 .n
n
nC
y x i i R x x a x x
The asymptotic relations (14.92)-(14.95) hold for (0, ), : 0 arg( ) 2.x
Taking stock, we have just identified (2) of our (4) missing asymptotic relations; 1( , )y x
near x and 3( , )y x near 0.x In addition, we have found the connection
coefficient 2 ( )C that is defined by the relation (12.23) as
(14.96) 1
22 1( ) ( ) .C i i R
147
Moreover we find by the relations (12.24) and (14.94) that
(14.97) 1
21 1( ) ( ) .K R i
We can now address one disadvantage that has stemmed from the rather abstract
definition for our canonical non-principal solution ( , )npcy x in equation (8.111), namely,
the constant constant ( )B that appears in relation (8.110) has proven to be resistant to
revealing a more explicit representation than what already appears in relation (8.110).
We will now pursue the 2 ( , )y x asymptotic relation near x and give also in parallel
the determination of the asymptotic relation for 4 ( , )y x near 0.x From Theorem 15,
it suffices to obtain the 2 ( , )y x asymptotic relation near x and the 4 ( , )y x
asymptotic relation near 0.x Following the same procedure that was successfully
employed to obtain the asymptotic relations for 1( , )y x near x and 3( , )y x near
0,x we start with the linear relations (ii)-(iii), in Theorem 12, and apply them towards
the functions (4) ( )g i x and (3) ( )g x to find
(14.99) (4) 1( ) exp 2 ( ) exp 2 ( ) ,
2 sing i x i g i x i g i x
i
and
(14.100)
(3) 1( ) exp 2 ( ) exp 2 ( ) .
2 sing x i g x i g x
i
148
Next we use the symmetry relation (14.5) to eliminate the functions ( ),g i x ( ),g i x
12( ),g x and
12( )g x that appear in relations (14.99)-(14.100). This gives
(14.101)
(4) 1 2 1 21( ) exp ( ) exp ( ) ,
2 sing i x i g x i g x
i
and
(14.102) (3) 1 2 1 21( ) ( ) ( ) .
2 sing x g i x g i x
i
Next we use (i) in Theorem 12 to eliminate the functions ( ),g x ( ),g x
( ),g i x and ( )g i x that appear in relations (14.101)-(14.102). This gives
(14.103) (4) (3) (4)
3 1( ) ( , ) ( ) ( , ) ( ),g i x R g x R g x
and
(14.104) (3) (3) (4)
1 2( ) ( , ) ( ) ( , ) ( ).g x R g i x R g i x
We clarify here that equation (14.99) is valid for ( , ) (0, )x , whereas equations
(14.100)-(14.104) are valid for (0, ), : 0 arg( ) 2.x From equations
(14.103)-(14.104) and the relations (14.64), (14.66) we then obtain the asymptotic
relations (14.105)-(14.108) for (0, ), : 0 arg( ) 2,x
(14.105) 1
(2) 23
0
( , ) ~ ( ) exp( ) , ,n
n
n
y x i i R i x a i x x
and
149
(14.106) 1
(4) 22
0
( , ) ~ ( ) exp(1 ) , 0 .n
n
n
y x i i R x x a x x
Next we employ (14.105), (14.94) and (14.106), (14.95) to give the asymptotic relations
for 2 ( , )y x near x and 4 ( , )y x near 0x for (0, ), : 0 arg( ) 2,x
(14.107)
2
1 12 2
2 3 1
0( )
( , ) ~ ( ) ( ) ( ) exp( ) , ,n
n
nK
y x i i R R i i x a i x x
and
(14.108) 1 1
2 24 2 1
0
( , ) ~ ( ) ( ) ( ) exp(1 ) , 0 .n
n
n
y x i i R i i R x x a x x
Summarizing, we have now obtained two previously unknown asymptotic relations for
our solutions 2 ( , )y x and 4 ( , )y x albeit these relations exhibit dependence on the
unknown contants ( ) and ( ) which received their definitions in Theorem 15.
Specifically, from (12.25) and (14.107), we have the ( ) dependent 2 ( )K
representation
(14.109) 1 1
2 22 3 1( ) ( ) ( ) ( ) .K i i R R i
We now seek to root out ( ) from our 2 ( )K representation (14.109). Observe that
from statement (i) in Theorem 15 and (12.17) we have (0, ), : Im 0,x
150
(14.110)
2
(2)
3 1 1 2 1
( , )
(2)
1 2 1 2
( , ) ( ) ( , ) ( ) 1 ( , ) ( ) ( , )
( ) ( ) ( ) ( , ) ( ) ( , ).
y x
y x C y x C y x y x
C C y x C y x
But we also have from (14.63)-(14.65), (14.89), Theorem 14, that
(0, ), : 0 arg 2,x
(14.111) 1 1
(2)2 23 3 1 1( , ) ( ) ( , ) ( ) ( , )y x i R y x i i R y x
Comparison of (14.110) and (14.111) reveals that our connection coefficients 2 ( )C and
1( )C satisfy the relations
(14.112) 1
22 1( ) ( )C i i R ,
and
(14.113) 1
21 3 2( ) ( ) ( ) ( ),C i R C
where we find here that (14.112) is in agreement with the relation (14.96) that we
obtained independently by use of solution asymptotics. It should now be clear from the
relations (14.109) and (14.113) that if any of 2 ( )K , ( ), or 1( )C is obtained, then all
three of these quantities will be known. Clearly possession of an independent equation to
(14.109) containing the unknowns ( ) and 2 ( )K or an independent equation to
(14.113) containing the unknowns 1( )C and ( ) would be sufficient so that a 2 2
system for two of these unknowns could be solved however, determination of such an
independent equation containing any two 2 ( )K , ( ), or 1( )C has, up to this point,
151
proven to be intractable. This may be a result of the fact that both of the connection
relations in (12.17)-(12.18) inherently contain ( , )npcy x with its atypical definition
received in (8.111) and its abstract definition which does not lend itself well to further
analysis. Therefore, we now pose the question, “Can we complete the solution of the
connection problem relating our three other solutions, 1( , )y x , 3( , )y x , and 4 ( , )y x ?”
We now address the connection problem, for (0, ), : Im 0.x
(14.114) 1 1 3 2 4( , ) ( ) ( , ) ( ) ( , )y x E y x E y x .
But we also have from (14.63)-(14.66), (14.89), (14.104), Theorem 14, for
(0, ), : 0 arg( ) 2.x
(14.115)
1 1(2)2 2
3 3 1 1
1 1(4) (2)2 2
1 1 2
( , ) ( ) ( , ) ( ) ( , )
( , ) ( ) ( , ) ( ) ( , )
y x i R y x i i R y x
y x i R y x i i R y x
.
Now substituting into (14.115) for (4) ( , )y x and
(2) ( , )y x by use of Theorem 15 gives
(0, ), : 0 arg( ) 2.x
(14.116)
1 1 12 2 2
3 3 1 1 1 2
1 1 12 2 2
4 3 1 2 1 2 2
( , ) ( ) ( ) ( , ) ( ) ( , )
( , ) ( , ) ( ) ( ) ( , ) ( ) ( , )
y x i R i i R y x i i R y x
y x y x i R i i R y x i i R y x
Next, we resolve the difficulty associated with 2 ( , )y x by performing elimination in
152
(14.116) to eliminate it and also find that the ( ) terms drop out. Solving for 1( , )y x
we obtain for (0, ), : 0 arg( ) 2,x
(14.117)
1 1 12 2 2
2 1 11 3 41 1 1 1
2 2 2 2
( ) ( ) ( ) ( )( , ) ( , ) ( , )
( ) ( ) ( ) ( )
i i R i i R i i Ry x y x y x
i i i i i i
.
Finally, comparing (14.117) with (14.114), we find the coefficients 1( )E and 2 ( )E as,
(14.118)
1 12 2
2 11 1 1
2 2
( ) ( ) ( )( ) ,
( ) ( )
i i R i i RE
i i i
and
(14.119)
12
12 1 1
2 2
( )( ) .
( ) ( )
i i RE
i i i
We find as to be expected that
(14.120) 1
22 1 1( ) ( ) ( ) .E K i R
Taking stock, we have identified another connection coefficient appearing in a linear
dependence relation between our solutions. Next we will give a determination of
constant ( ) appearing in (14.118) and thereby obtain an explicit representation of the
connection coefficient 1( )E . From its definition in Theorem 15, we have
(14.121) (4)
4
3
( , ) ( , )( ) , (0, ), : 0 arg 2.
( , )
y x y xx
y x
We evaluate (14.121) at a fixed x value 0,x c obtaining
(14.122) (4)
4
3
( , ) ( , )( ) , : 0 arg 2.
( , )
y c y c
y c
153
Thus from (14.122) we find the connection coefficient 1( )E as
(14.123)
(4)1 1
42 22 1
3
1 1 12 2
( , ) ( , )( ) ( )
( , )( ) ,
( ) ( )
y c y ci i R i i R
y cE
i i i
Taking stock once more, with the determination of 1( )E and 2 ( )E , we have now
solved the connection problem (14.114). One question that now arises is whether or not a
similar approach could be used to find ( )? From Theorem 14, statement (i), it
follows that,
(14.124) (2)
2
1
( , ) ( , )( ) , ( , ) 0, .
( , )
y x y xx
y x
In our next investigation, we pose the question, can we solve completely the connection
problem relating 3( , )y x to the fundamental system near 0x consisting of 1( , )y x
and ( , )y x ? Note that in this connection relation, we have swapped ( , )npcy x with the
non-principal solution ( , )y x given by relations (8.87) & (8.88). Thus we now consider
the connection relation for (0, ), : Im 0,x
(14.125) 3 13 1 14( , ) ( ) ( , ) ( ) ( , ).y x E y x E y x .
We start by giving a subdominance argument to prove the following theorem.
Theorem 16: For the solutions given in (8.87)-(8.88), (14.64), and (8.97),
(i) (2)
1( , ) 1 ( , ) ( ) ( , )y x y x y x , (0, ), .x
Here is an unknown constant.
154
PROOF: Since (2)
1( , ), ( )y x y x is a fundamental system to (8.5) near 0,x our
( , )y x solution defined near 0x must be a linear combination of these solutions or
(14.126) (2)
1( , ) ( ) ( , ) ( ) ( , ),y x y x y x (0, ), .x
where ( ) and ( ) are constants. The determination that 1 comes from the fact
that ( , )y x and (2) ( , )y x have identical asymptotic relations near 0.x Observe that
if 1 , then (14.126) contradicts the fact that the asymptotics near 0x for ( , )y x
and (2) ( , )y x match identically so 1. Unfortunately, ( ) in relation (14.126)
cannot be directly determined by use of asymptotics near 0x as the term
1
1( ) ( , ) ( )xy x O xe
and hence it tends rapidly to zero as 0 .x We have thus
proven relation (i) and the theorem is established.
Next, we use Theorem 16 to substitute into (14.125) for 2 ( , )y x and compare with
(14.111) to conclude
(14.127) 1
213 3 14( ) ( ),E i R E
12
14 1( ) .E i i R
As expected we find 14 2( ).E C Thus one of our connection coefficients, 14E , is now
known. So as to identify 13E , we now need only an explicit representation of ( ).
We first observe from Theorem 16 the asymptotic relation for ( , )y x near x for
(0, ), : 0 arg( ) 2,x is give by
155
(14.128)
1 1
32 23 1
( )
( , ) ~ ( ) ( ) ( ) exp( ) 1 ( ) , .
K
y x i i R R i i x O x x
Now to find ( ), ( )K , and 13E , we solve for ( ), in Theorem 16 to obtain
(14.129) (2)
1
( , ) ( , )( ) , 0, , .
( , )
y x y xx
y x
We evaluate at a fixed x value 0, ,x c to obtain for this choice of x
(14.130) (2)
1
( , ) ( , )( )
( , )
y c y c
y c
Thus from (14.127) we find explicitly the connection coefficient 13E as
(14.131)
(2)1
213 3 14
1
( , ) ( , )( ) .
( , )
y c y cE i R E
y c
Furthermore, we find from (14.128) that
(14.132)
(2)1 1
2 23 1
1
( , ) ( , )( ) ( ) ( )
( , )
y c y cK i i R R i
y c
Taking stock once more, with the determination of 13E and 14E , we have now solved the
connection problem (14.125).
Similar to the determination of ( ) in (14.130), we may observe that any choice of
(0, )x in (14.124) can be used to make a definition of ( ). Therefore by choosing
a fixed x value ,x c we may take ( ) as
156
(14.133) (2)
2
1
( , ) ( , )( ) , .
( , )
y c y c
y c
By this choice for ( ) and use of Theorem 15 (i), substitution into (14.116) gives
(14.134) 3 1 1 2 2( , ) ( ) ( , ) ( ) ( , ),y x C y x C y x
where
(14.135)
(2)
1 1 122 2 2
1 3 1 2 1
1
( , ) ( , )( ) ( ) ( ) , ( ) ( ) .
( , )
y c y cC i R i i R C i i R
y c
Here equations (14.134), (14.125), and (14.115) come in the eigenfunction expansion
theory for (8.5) on (0, );x namely, the Titchmarsh-Weyl m function will be a
quotient of 1( )C and 2 ( ).C For this reason, we should expect that for in the 1st
quadrant, the branch of ( )m in this m function should be placed on the positive real
axis, similarly to the Frobenius problems treated in Fulton [48] and Fulton and
Langer [42]. For the equation (8.5) with the singular endpoints 0x and x of
Poincaré rank 1, the relations of linear dependence given in (14.125), (14.115), and
(14.134) represent solutions of the “Sturm-Liouville Connection Problem”, as formulated
by C. Fulton in [47]. That is, given a Sturm-Liouville equation
(14.136) ( ) , ,py qy ry a x b
with
(I) x a being a singular endpoint of type LC/N or LP/N,
157
and
(II) x b being a singular endpoint of type LC/N, LP/N, or LP/O-N,
let ( , ), ( , ) be a fundamental system of solutions normalized at x a so as to
satisfy all of the properties
(i) Both and are entire in for fixed ( , ),x a b
(ii) 2 0( , );L a x r for 0 ( , ),x a b
(iii) Both and are real-valued for ( , )x a b and ( , ),
and
(iv) ( ) ( , ) 1.xp x W
Let ( , )x be the solution which is square-integrable at the LP endpoint .x b Then
the Sturm-Liouville Connection Problem is the problem of finding the (complex-valued)
constants 1( )C and 2 ( )C in the relation of linear dependence ( ( , ),x a b
: Im 0),
(14.137) 1 2( , ) ( ) ( , ) ( , ) ( , ).x C x C x x
The Titchmarsh-Weyl m function arises from this relation of linear dependence as
(14.138) 1 2( ) ( ) ( ).m C C
The solution ( , )x corresponds to 3( , )y x in (14.125), (14.115), and (14.134), and
the fundamental systems (2)
1 1, , , ,y y y y and 1 2,y y correspond to possible choices
158
of the fundamental system , normalized near 0.x In the next section, we briefly
discuss the determination of the characteristic exponents.
159
Section 15: Determination of the Characteristic Exponents:
We’ll now address the determination of the characteristic index parameter appearing in
our connection coefficients. Depending on the context, this parameter has
several names in the literature including a characteristic index, characteristic exponent,
circuit exponent, and Floquet exponent [24]. In the context of the Mathieu equation,
Whittaker and Watson’s book gives a historical account that G. W. Hill, in 1886, in his
investigations of Lunar Theory, was the first to offer an analytic solution towards
obtaining such a characteristic exponent by means of “infinite determinants” which are
now referred to in the literature as Hill determinants [104]. In the context of the Mathieu
equation from [104], [75]-[76], [28], [33], [80], [30]-[34], and [92] we find that the
characteristic exponent is a function of the parameters and 2h appearing in
the Mathieu equation and Modified Mathieu equation
(15.1) 2
2
22 cos(2 ) 0
dh z
dz
, Mathieu equation
or
(15.2) 2
2
22 cosh(2 ) 0
dh z
dz
. Modified Mathieu equation
Moreover, after a survey of the literature we find that our SL equation (8.5) as well as
Bühring’s (14.2) can be brought into the forms (15.1) and (15.2) above by means of
suitable changes of variables. See for instance [80] or [92]. Following the analysis in
[80, p. 437] for instance, we can apply a series of transformations to convert our SL
equation (8.5) into either of (15.1) or (15.2) after which the classical treatments on the
160
Mathieu equation in [75]-[76], [28], [33] would be applicable. Specifically, following the
analysis in [80, p. 437], we find that with the changes of dependent variable, independent
variable, and parameters in (8.5),
(15.3) 1
2y x , zx e ,
14exp( 4)i
, 1
4exp( 4)h i ,
our SL equation (8.5) is transformed into the modified Mathieu equation (15.2) with the
parameter values 1 4 and 1
2 2h i . From my further investigations, I encountered
several numerical routines and approximation formulae that have been employed by prior
authors to generate asymptotic representations for the characteristic exponents
specifically in the context of Floquet solutions to (15.1) and (15.2) where the
characteristic exponents are given (approximately) as functions of and 2h and the
schemes simplify for either when 2h is small or when
2h is large. See for instance [55],
[80], [87], and [79]. Following the analysis from either ([92], p. 1186-1189), ([80, p.
339-344]), or [79, p. 556-559], when 2h is small (that is small in our SL equation
(8.5)), the characteristic exponents may be approximated as 1
2 1 2 .
More generally in the literature, the characteristic exponents are approximated by either
perturbation methods as given in [80, p. 345, eq. 17.9a], by use of continued fractions as
given in [79, p. 559], or through treatments involving transcendental equations as given
in [92, p. 1189]. For instance, the Dingle- Müller approximation scheme from [80, pg.
155] yields the approximation for the characteristic exponents as
161
(15.4)
2 8412
1 2 3 3 2
8 35 151 2 ( )
4 1 64( 4)( 1)
hhO h
.
Using this scheme, we find that our characteristic exponents may thus be approximated as
(15.5) 2 31 2 2( )
2 3 135O .
Further methods to evaluate the Hill determinant numerically have been investigated in
[77]-[78], [99]-[100], and [24] though we will defer such numerical investigations to the
numerical mathematicians. In the next section, we make some considerations of the more
general 1Bx potential SL equation.
162
Section 16: Some Brief Considerations on the 1Bx Potential
Here we conjecture a method of attack towards the more general 1Bx potential SL
equation by use of the methods developed in the analysis of the 41 x potential SL
equation in Sections 8-12, 14. Consider the SL equation (16.1) where , 2,B B
(16.1) 1
, 0 ,B
y y y xx
.
When 0, : 1,B B H.L. Turrittin gives in [98, p. 304] analysis of this equation.
Near the LP irregular singular point of finite rank 0,x we may expect that
independent controlling factors for two linearly independent solutions to (16.1) can
be obtained from the 0 case likewise to what occurred for when 4.B Taking
0 in (16.1) & employing the changes of variables
(16.2) 1
2( ) ( )y x x Y x and 2
22
2
B
t xB
transforms (16.1) into the modified Bessel equation of order 1
2B
,
(16.3)
2 2
2
1( ) ( ) ( ) 0
2t Y t tY t t Y t
B
.
Hence for this equation, with the additional restriction that 2 1B n , n , 0n , we
find the standard fundamental system to (16.3) from [84] as
(16.4) 1 2
( ) ( )( ) ( ) , ( ) ( ) .
2 sin( )
I t I tY t K t Y t I t
163
Here the modified Bessel function ( )I t has the series representation
(16.5) 2
0
.251( ) .
2 ! ( 1)
k
n
k
tI t t
k k
It follows that for when 0 a fundamental system to (16.1) is
(16.6) 1 1
2 21 2( ) ( ( )), ( ) ( ( )) .y x x K t x y x x I t x
Now by analogy to what occurred in the case when 4,B we conjecture that when in
(16.1) is left unrestricted, two linearly independent solutions to (16.1) will have the form
(16.7) 1
2( , ) ( ) ( , )p By x x K t x w x , 1
2( , ) ( ) ( , ).np By x x I t x v x
Here the asymptotics for K and I are well known and may be found as given in [84,
252]. These asymptotics motivate the designation of principal and non-principal
solutions. We may further anticipate that ( , )Bw x and ( , )Bv x will have asymptotic
representations as given by the classical formal power series method,
(16.8) 1 1
( , ) ~ 1 ( , ) , ( , ) ~ 1 ( , ) .k k
B k B k
k k
w x a x x v x b x x
To Be Done: Near 0,x so as to localize a principal solution ( , )py x and canonical
choice for a non-principal solution ( , )npcy x , as was done with the 41 x potential SL
equation, we conjecture that the insertion of the changes of variables in (16.7) into (16.1)
will give rise to ODEs for the dependent factors ( , )Bw x and ( , )Bv x . We may
further conjecture that by further imposing suitable ICs at 0x and converting such a
164
( , )Bw x IVP into its equivalent Volterra integral equation as was done with the 41 x
potential SL equation, proofs along the lines of those given in Sections 8-11 could yield
the following properties of solutions ( , )py x and ( , )npcy x (See [49] and [51].).
(i) ( , )py x is uniquely defined by suitable ICs, square-integrable near 0,x
( , ) ( , )p py x y x , and entire in .
We then hypothesize that from the reduction of order technique, using such a uniquely
defined ( , )py x , as was shown in Sections 8-11 in the context of the 41 x potential SL
equation, we could localize a canonical choice for the non-principal solution near 0.x
We anticipate that such a ( , )npcy x will have the the properties
(ii) ( , )npcy x is uniquely defined by the unique choice of ( , )py x ,
( , ) ( , )npc npcy x y x , and entire in .
Futhermore, analysis in parallel to our treatment for the irregular singular point at x
with regards to the 41 x potential SL equation in Sections 8-11 may likewise yield
analogues of the solutions 3( , )y x and 4 ( , )y x for which asymptotics could be
determined by means of integration by parts as applied to suitable Volterra integral
equations OR by the application of the theory of formal power series solutions. The
controlling factors of these solutions should work out from the theory of formal power
series in Jörgens [67].
165
Summary of Results:
In this chapter, new methods were presented enabling singular behavior of solutions of
SL equations with strongly singular potentials to be factored away yielding Volterra
integral equations for their remaining dependent factors. The methods were carried
out for the 2nd order Sturm-Liouville equation with strongly singular 41 x potential and
with irregular singular points at 0x and .x Representations of solutions were
thereby obtained, generated about the irregular singular points, 0x and ,x and the
analysis given to establish the existence and uniqueness results obtained in this chapter
are of interest in that they do not rely on the standard theorems as the standard theorems
were not applicable near 0x and .x In addition, the determination of a principle
and non-principal solution was achieved near the irregular singular point 0x by novel
means, namely, through the combination of substitutions so as to factor away highly
singular independent controlling factors and then by the use of reduction of order and
Volterra integral equations of the 2nd kind to formulate the remaining dependent
factors by recursion. Specifically of interest is the imposition of intitial conditions at the
irregular singular point to define a principal solution, an approach that appears to be
nowhere else in the literature. Several connection problems too were addressed and this
work should inform all future investigators in this area of both the pros and cons of
formulating a canonical choice for a non-principal solution near an LP singular endpoint
in regards to associated connection problems. Some work towards generalizations with
the 1Bx was also discussed.
166
Chapter 3: New Methods in Spectral Theory
A New Calculation of the Spectral Density Function for Bessel’s Equation
Chapter 3 Introduction: In the paper by C. Fulton, D. Pearson, and S. Pruess [45], a
new characterization of the spectral density function ( )f is given for a Sturm-Liouville
equation. This method makes use of the Appell System, a companion linear system of
ordinary differential equations. Here original work by this author demonstrates the first
nontrivial example of a spectral density function calculation using this new technique.
Section 17: Background: The Titchmarsh-Weyl m-Function
Consider the general Sturm-Liouville equation:
(17.1) 1( ) ( ) ( ) ( ), ( , ), ( ) ( , ),y x q x y x y x x a q x L a and lim ( ) 0.x
q x
Suppose x a is a singular endpoint of the Limit Point (LP) singular endpoint
classification.
Let ( , ), ( , )u x v x be the fundamental system of (16.1) for which
( , ), ( , ) 1aw u x v x , so that
(17.2) ( , ) ( , ) 1 0
( , ) ( , ) 0 1
u a v a
u a v a
.
Then the Titchmarsh-Weyl m function, ( ),m is defined by the requirement
(17.3) 2( , ) ( , ) ( ) ( , ) ( , ).x u x m v x L a
167
As (17.1) is a linear differential equation, ( , )x is necessarily a solution to (17.1) near
x a . Moreover, due to x a being an LP singular endpoint, ( , )x and ( )m are
uniquely defined in (17.3) by this square-integrability requirement [95, p. 86].
The spectral density function ( )f is then characterized by the Titchmarsh-Kodaira
formula:
(17.4)
0
Im ( )( )( ) lim
m iKf
,
where
(17.5) 0
0
( ) lim Im ( )K m i d
.
See [95, p. 54, eq. 3.3.1]. In the next section, we define the Appell system.
168
Section 18: The Appell System
In 1880, M. Appell gave a companion system to the Sturm-Liouville equation (17.1) as
(18.1)
( ) 0 ( ) 0 ( )
( ) ( ) 2 0 2( ( )) ( )
( ) 0 1 0 ( )
P x q x P xdU d
x Q x q x Q xdx dx
R x R x
.
See [6]. Some fundamental properties of (18.1) are now given with select PROOFs by
this author:
(i) If ( )y x is any solution of the Sturm-Liouville equation (17.1), then
2
2
( )
( ) 2 ( ) ( )
( )
y x
U x y x y x
y x
is a solution to the Appell System (18.1).
PROOF: Suppose ( )y x is any solution of the Sturm-Liouville equation (17.1). Then
2
2
2
2 2
2 ( ) ( )( )
( ) 2 ( ) ( ) 2 ( ) ( ) ( )
( ) 2 ( ) ( )
( ) ( 2 ( ) ( ))
2 ( ) 2( ( )) ( ) ( ).
2 ( ) ( )
y x y xy xdU d
x y x y x y x y x y xdx dx
y x y x y x
q x y x y x
dUy x q x y x x
dxy x y x
Here we have used the relation ( ) ( ) ( ).y x q x y x
(ii) Let ( , ), ( , )u x v x be the fundamental system of (17.1), where (17.2) holds.
169
Then a fundamental system of three linearly independent solutions to (18.1) is
2 2
2 2
( ) ( ) ( ) ( )
2 ( ) ( ) , ( ) ( ) ( ) ( ) , 2 ( ) ( )
( ) ( ) ( ) ( )
u x u x v x v x
u x u x u x v x u x v x v x v x
u x u x v x v x
.
PROOF: Observe that by property (i), column vectors one and three are already known
known to be linearly independent solutions of Appell System (18.1). Thus it must only
be established that column vector two also satisfies Appell System (18.1).
Let
( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( )
u x v x
U x u x v x u x v x
u x v x
.
Now utilizing ( ) ( ) ( )u x q x u x and ( ) ( ) ( )v x q x v x , it follows that
( ) ( 1)[ ( ) ( ) ( ) ( )]
( ) 2 ( ) ( ) 2( ( )) ( ) ( ) ( ) ,
( ) ( ) ( ) ( )
q x u x v x u x v xdU dU
x u x v x q x u x v x xdx dx
u x v x u x v x
and thus column vector two has been established as a solution to Appell System (18.1).
(iii) For any solution
( )
( ) ( )
( )
P x
U x Q x
R x
to Appell System (18.1), let the indefinite inner
product be defined by: 1 2 1 2 2 1 1 2( ), ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ).U x U x P x R x P x R x Q x Q x
Then we have:
( ) ( ) ( ) ( )
( ) ( ) ( ) 2 ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
u x v x u x v xdU
x u x v x u x v x u x v xdx
u x v x u x v x
170
2( ), ( ) 4 ( ) ( ) ( )U x U x P x R x Q x const .
PROOF:
2( ), ( ) (4 ( ) ( ) ( )) 4 ( ) ( ) 4 ( ) ( ) 2 ( ) ( ).d d
U x U x P x R x Q x P x R x P x R x Q x Q xdx dx
Now using ( ) ( ) ( )P x q x Q x , ( ) 2 ( ) 2( ( )) ( ),Q x P x q x R x ( ) ( )R x Q x ,
we find ( ), ( ) 0d
U x U xdx
2( ), ( ) 4 ( ) ( ) ( )U x U x P x R x Q x const .
(iv) Let
1
1 1
1
( )
( ) ( )
( )
P x
U x Q x
R x
and
2
2 2
2
( )
( ) ( )
( )
P x
U x Q x
R x
be any two solutions to Appell System
(18.1) where
2 2
1 1 1 1
2 2
( ) ( ) ( ) ( )
( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( )
( ) ( ) ( ) ( )
u x u x v x v x
U x a u x u x b u x v x u x v x c v x v x
u x u x v x v x
,
2 2
2 2 2 2
2 2
( ) ( ) ( ) ( )
( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( )
( ) ( ) ( ) ( )
u x u x v x v x
U x a u x u x b u x v x u x v x c v x v x
u x u x v x v x
.
Then
1 2 1 2 1 2 1 2( ), ( ) 2( )U x U x a c c a bb and 2
1 1 1 1 1( ), ( ) 4U x U x a c b .
171
(v) Let ( )y x be any solution to the Sturm-Liouville equation (17.1) and let
( ) ( ) ( ) ( )T
U x P x Q x R x be a solution to Appell System (18.1). Then
22( ) ( ) ( ) ( ) ( ) ( ) ( ) 0
dP x y x Q x y x y x R x y x
dx
.
(vi) Near x , 0 0x , if either 1 0( ) ( , )q x L x or 1 0( ) ( , )q x L x ,
0( ) ( , )locq x AC x , and lim ( ) 0x
q x
, then the terminal value problem below has a
unique solution.
(18.2)
( ) 0 ( ) 0 ( )
( ) ( ) 2 0 2( ( )) ( )
( ) 0 1 0 ( )
( )
lim ( ) 0 , 0,
( ) 1/
x
P x q x P xdU d
x Q x q x Q xdx dx
R x R x
P x
Q x
R x
.
See [46].
(vii) Let
1
1 1
1
( )
( )
( )
P x
U Q x
R x
be the unique solution to the terminal value problem (18.2).
Then necessarily 24 4ac b . Furthermore, when 0x is either regular or a RSP of
LC/N or LP/N type, x is LP/O-N with cut-off 0, and ( )q x is absolutely
integrable near ,x the spectral density function, ( ),f for (0, ), is
characterized by:
172
(18.3)
22
1 1 1
1 1( )
( ) ( , ) ( ) ( , ) ( , ) ( ) ( , )f
a P x v x Q x v x v x R x v x
,
where ( )f is absolutely continuous for (0, ) (See [45] or [47, p. 40].).
In the next section, we’ll utilize these properties and demonstrate the viability of the
spectral density function characterization given in (vii) by providing the first nontrivial
example of a spectral density function calculation by use of (18.3), as applied to the
Bessel equation in Liouville-Normal form.
173
Section 19: Calculation of the SDF for Bessel’s Equation in Liouville-Normal Form.
Consider the Bessel Equation in Liouville-Normal form
(19.1) 2
2
1 4( ) ( ) ( ),y x y x y x
x
for ,a x 0a and with 0,1,2, .
Here observe that x is a LP/O-N singular endpoint with cutoff 0.
Let ( , ), ( , )u x v x be the fundamental system to (19.1) such that
( , ) ( , ) 1 0
( , ) ( , ) 0 1
u a v a
u a v a
, for all 0, . Here the potential function
2
12
14( ) ( , )q x L a
x
and thus the Appell System property (vi), from above, holds.
The corresponding Appell System terminal value problem for (19.1) is then
(19.2)
2
2
2
2
1 40 0
( ) ( )2 1 2
( ) 2 0 2 ( )
( ) ( )
0 1 0
( )
lim ( ) 0 , 0,
( ) 1x
xP x P x
dU dQ x Q x
dx dx xR x R x
P x
Q x
R x
.
Let 1( )U x be the unique solution to (19.2) where a , b , and c are defined by the relation
174
(19.3)
2 2
1 1 1 1
2 2
( ) ( ) ( ) ( )
( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( )
( ) ( ) ( ) ( )
u x u x v x v x
U x a u x u x b u x v x u x v x c v x v x
u x u x v x v x
.
Note here that the fundamental system ( , ), ( , )u x v x to (19.1) satisfying the
Wronskian requirement ( , ), ( , ) 1aw u x v x is uniquely determined and it may be
explicitly written with
(19.4) 1 2( , ) ,v x C xJ x C xY x
(19.5) 3 4( , ) ,u x C xJ x C xY x
where the constants 1 2 3, , ,C C C and 4C are given by
(19.6) 1 2( ), ( ),2 2
a aC Y a C J a
3 ( ) ( ),24
C Y a a Y aa
and
4 ( ) ( )24
C J a a J aa
.
The definitions for ( , ), ( , )u x v x , 1 2 3, , ,C C C and 4C ensure ( , ), ( , ) 1aw u x v x ,
as verified by application of Wronskian relations for the Bessel functions ( )J x and
( )Y x (See [101, p. 76].). Now that the fundamental system to (19.1) has been given
explicitly, we can use the characterization of solution as given in (19.3) and impose the
terminal condition in (19.2) to yield
175
(19.7)
2 2
2 2
lim ( ( , )) ( , ) ( , ) ( ( , ))
0 lim ( 2 ( , ) ( , ) [( 1)( ( , ) ( , ) ( , ) ( , )] 2 ( , ) ( , )
1 lim ( , ) ( , ) ( , ) ( , )
x
x
x
a u x bu x v x c v x
a u x u x b u x v x u x v x c v x v x
au x bu x v x cv x
,
where a , b , and c are uniquely defined by (19.7). For further progress towards
obtaining explicit representations of a , b , c and then ( )f as characterized by (18.3), we
next make use of known asymptotic relations for the Bessel functions ( )J x and ( )Y x
as x :
(19.8) 1
22 1
( ) cos2 4
x J x x Ox
,
(19.9) 1
22 1
( ) sin2 4
x Y x x Ox
,
(19.10) 1
22 1
( ) sin2 4
x J x x Ox
,
(19.11) 1
22 1
( ) cos2 4
x Y x x Ox
.
By letting ( , ) ( ) :2 4
w x w x x
and applying asymptotic relations (19.8)-
(19.11) to (19.7) we obtain
176
(19.12)
2 2 2
3 1 3 1
2 2 2
4 2 4 2
3 4 2 3 1 4 1 2
sin ( ( )) ( ) ( )
2 1lim cos ( ( )) ( ) ( )
sin( ( ))cos( ( )) 2 ( ) 2
x
w x a C bC C c C
w x a C bC C c C Ox
w x w x aC C b C C C C cC C
,
(19.13)
2
3 4 2 3 1 4 1 2
2
3 4 2 3 1 4 1 2
2 2 2 2
4 3 2 4 1 3 2 1
sin ( ( )) 2 ( ) 2
2 10 lim cos ( ( )) 2 ( ) 2
sin( ( ))cos( ( )) 2 (( ) ( ) ) 2 ( ) 2 (( ) ( ) )
x
w x aC C b C C C C cC C
w x aC C b C C C C cC C Ox
w x w x a C C b C C C C c C C
,
and
(19.14)
2 2 2
4 2 4 2
2 2 2
3 1 3 1
3 4 2 3 1 4 1 2
sin ( ( )) ( ) ( )
1 2 1lim cos ( ( )) ( ) ( )
sin( ( ))cos( ( )) 2 ( ) 2
x
w x a C bC C c C
w x a C bC C c C Ox
w x w x aC C b C C C C cC C
.
Now to satisfy relations (19.2)-(19.4), nine equations emerge, three of which are
independent,
(19.15)
2 2
3 1 3 1
2 2
4 2 4 2
3 4 1 4 2 3 1 2
( ) ( )2
( ) ( )2
2 ( ) 2 0
a C bC C c C
a C bC C c C
aC C b C C C C cC C
,
or in a matrix form
177
(19.16)
2 2
3 1 3 1
2 2
4 2 4 2
3 4 1 4 2 3 1 2
2( ) ( )
( ) ( )2
2 ( ) 2 0
aC C C C
C C C C b
C C C C C C C C c
.
The solution to (19.16) is computed to be
(19.17) 2 2
1 2
2
2 3 1 4
( ) ( )
2( )
C Ca
C C C C
,
1 3 2 4
2
2 3 1 4( )
C C C Cb
C C C C
, and
2 2
3 4
2
2 3 1 4
( ) ( )
2( )
C Cc
C C C C
.
Futher computation with 1 2 3, , ,C C C and 4C as defined in (19.6) reveals
(19.18) 2 2( ) ( )2
aa J a Y a
,
(19.19)
2 2( ) ( ) ( ) ( ) ( ) ( )2
v v v vb J a Y a a J a J a Y a Y a
,
(19.20)
Finally, we may apply the characterization of the spectral density function as given in
(18.3) obtaining
(19.21) 2 2 2
2 1( )
( ) ( )f
a J a Y a
.
In the next brief section, we’ll validate the results obtained here using some independent
checks.
2 22 2( ) ( ) ( ) ( )
8 2
( ) ( ) ( ) ( )2
v v v v
ac J a Y a J a Y a
a
J a J a Y a Y a
178
Section 20: Validation of Results
(A) To confirm the validity of the representations a , b , c , and ( )f , we first compare
the spectral density function obtained in (19.21) with the spectral density function for the
Bessel Equation (18.1) obtained via the Titchmarsh-Kodaira formula (17.4). See [95, p.
86]. Here we find full agreement. Thus we may conclude that the representations
obtained in (19.18) and (19.21) for a and ( )f have been validated.
(B) According to C. Fulton, D. Pearson, and S. Pruess [45], the Titchmarsh-Weyl m-
function takes the form
(20.1) 1
( )2
bm i
a a ,
while according to Titchmarsh in [95], the m-function for Bessel Equation (19.1) is
(20.2) ( ) ( ) 1
( )2( ) ( )
J a iY am
aJ a iY a
.
By calculation of ( )m via (20.1) using (19.8-19.9), we indeed obtain (20.2) and thus we
may conclude that the calculation of b is validated.
(C) According to Appell System property (vii), necessarily 24 4ac b . Calculation of
the left-hand side, via application of multiple identities pertaining to Bessel Functions,
indeed returns 4. See particularly the Wronskian identities for the Bessel functions in
Watson [30, p. 76]. Thus we may conclude that the calculation of c is validated.
179
Summary of Results:
In this chapter, an original calculation of the spectral density function was performed
using the new method of C. Fulton, D. Pearson, and S. Pruess [45]. Prior to this work, no
other nontrivial example of such a calculation using the characterization of ( )f given in
(18.3) had been demonstrated. Future authors may follow the prescribed outline in this
dissertation to calculate additional spectral density functions for Sturm-Liouville
equations of the form (17.1)-(17.2).
Conclusion:
In this dissertation, new methods and results were obtained for Sturm-Liouville equations
with singular endpoints of Poincaré rank zero or one. Many of the methods developed
here seem to extend naturally to generalizations of the problems addressed in this
dissertation and thus it is the hope of this author that they will offer a source of progress
for future work in the dissertation topics by this author and others.
180
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