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Pochhammer symbol
5. Hypergeometric Functions
1 1 0x x y c a b x y ab y Hypergeometric equation( Gauss’ ODE & functions )
Regular singularities at 0,1,x
Solution : 2 1
0
, ; ;!
nn n
n n
a b xy F a b c x
c n
Hypergeometric function(series)
For a, b, c real, range of convergence are :
1 ,1
1 ,1 1
x c a b
x a b c a b
Series diverges for 1a b c
21 11
1 2!
a a b bab xx
c c c
1 1n
a a a a n
01a
Chap 7:
0, 1, 2,c
2 F1 includes many elementary functions.
E.g.
Sum terminates if 2 F1 = polynomial
Properties
2 10
, ; ;!
nn n
n n
a b xF a b c x
c n
0, 1, 2,c
or 0, 1, 2,a b
1
1
ln 1n
n
n
xx
n
0 1
n
n
xx
n
0
1 1
2 !
n
n n
n n
xx
n
2 1ln 1 1,1 ;2 ;x x F x
2nd Solution, Alternative ODE
1 1 0x x y c a b x y ab y
2nd Solution :
2 1
0
, ; ;!
nn n
n n
a b xF a b c x
c n
§ 7.6 :
12 1 1 , 1 ; 2 ;cy x F a c b c c x 2,3,4,c
c = integer y not independent of 2 F1 ( a, b ; c ; x)
additional logarithm term required
Alternative ODE :
2 1 1 11 1 1 2 0
2 2 2
z z zz y a b z a b c y ab y
2
2 2 2 22
1 21 2 2 1 4 0
d c dz y z a b z y z ab y z
d z z d z
11
2x z
11
2y x z y z
0, 1, 2,c
Contiguous Function Relations
2 1 2 1 2 11 , ; ; 1, 1 ; ; 1, 1 ; ;x F a b c x F a b c x F a b c x
2 21 1a b a b c a b
21a b a b
1c a a b b
1c b a b a
2 1 2 1 2 12 , ; ; 1 1, ; ; 1, ; ;a c b a x F a b c x a x F a b c x c a F a b c x
Hypergeometric Representations
2 1
2 ! 1, 2 1;1 ; 1
2 ! 1 2n
nC x F n n x
n
2 1
1, 1 ;1 ; 1
2nP x F n n x
/22
2 1
1! 1, 1 ; 1 ; 1
! 2 ! 2
m
mn m
xn mP x F m n m n m x
n m m
22 2 12
2 ! 1 1, ; ;
2 ! ! 2 2n
n n
nP x F n n x
n n
22 1
2 1 !! 1 1, ; ;
2 !! 2 2n n
F n n xn
22 1 2 12
2 1 ! 3 3, ; ;
2 ! ! 2 2n
n n
nP x F n n x
n n
22 1
2 1 !! 3 3, ; ;
2 !! 2 2n n
F n n xn
2 1
1 1, ; ; 1
2 2nT x F n n x 2 1
3 11 , 2 ; ; 1
2 2nU x n F n n x
22 1
3 11 1, 1 ; ; 1
2 2nV x n x F n n x
2 1
0
, ; ;!
nn n
n n
a b xF a b c x
c n
6. Confluent Hypergeometric Functions
0x y c x y a y Confluent Hypergeometric eq.
Singularities : regular at irregular at0x
Solution : 1 1
0
; ;!
nn
n n
a xF a c x
c n
For a, c real, series converge for all finite x .
211
1 2!
a aa xx
c c c
0, 1, 2,c
x
, ,y M a c x
Sum terminates if 1 F1 = polynomial0, 1, 2,a
E.g. 2
0
2x
terf x d t e
1
0
,
x
t aa x d t e t Re 0a
22 1 3, ,
2 2x M x
1, 1 ,ax M a a x
a
2nd solution :
0x y c x y a y
1 1 , 2 ,cy x M a c c x 2,3,4,c
Standard form :
1, , 1 , 2 ,, ,
sin 1cM a c x M a c c x
y U a c x xc a c c a c
Alternate ODE :
2
2 2 22
2 12 4 0
d c dy x x y z a y z
d x x d z
Integral Representations
1
11
0
, , 1c ax t ac
M a c x d t e t ta c a
0c a
11
0
1, , 1
c ax t aU a c x d t e t ta
Re 0 , 0x a
Techniques for verifying integral representations :
1. g(x,t) or Rodrigues relations.
2. Expand integrand into series & integrate.
3. (a) As solution to ODE. (b) Check normalization.
Confluent Hypergeometric Representations
1
, 2 1, 21 2 2
i xe xJ x M ix
1
, 2 1, 21 2 2
xe xI x M x
22
2 ! 1, ,
! 2n
n
nH x M n x
n
22 1
2 2 1 ! 3, ,
! 2n
n
nH x x M n x
n
, 1,nL x M n x
m
mmn n mm
dL x L x
d x !
, 1,! !
n mM n m x
n m
Further Observations
Advantages for using the (confluent) hypergeometric representations :
1.Asymptotic behavior or normalization easier to evaluate via the integral
representation of M & U.
2.Inter-relationship between special functions becomes clearer.
Self-adjoint version :
/2 1/2 1, 2 1,
2x
kM x e x M k x
Whittaker function
Self-adjoint ODE :
2
2
11 4 04k k
kM M
x x
2nd solution : /2 1/2 1, 2 1,
2x
kW x e x U k x
7. Dilogarithm
2
0
ln 1z
tLi z d t
t
Dilogarithm
Usage :
1.Matrix elements in few-body problems in atomic physics.
2.Perturbation terms in electrodynamics.
Expansion 2
0
ln 1z
tLi z d t
t
2
0
ln 1z
tLi z d t
t
1
10
zn
n
td t
n
21
n
n
z
n
1
0
z
pp
Li tLi z d t
t Poly-logarithm
1
n
pn
z
n
12
21
0 0
z zn
n
Li t td t d t
t n
31
n
n
z
n
3Li z
11
n
n
zLi z
n
01
n
n
Li z z
ln 1 z
11
1 z
0
1
1
z
d tt
1
z
z
For series converges & is real.
Analytic Properties 2
0
ln 1z
tLi z d t
t
2
1
n
n
z
n
Branch point at z = 1.
Conventional choice : Branch cut from z = 1 to z = .
with principal value : 2 21
n
n
zLi z
n
& 1z x x
For series diverges
but integral is finite & complex ( analytic continued ).
& 1z x x
For series diverges
but integral is finite & real ( analytic continued ).
& 1z x x
Mathematica
Since the only pole is at z = 0, the integral is
independent of path as long as it does not
cross the branch cut.
For the path colored blue in figure,
2
0
ln 1z
tLi z d t
t
1
20
1
ln 1ln 1 lnlim
1i
z
ii
e
t ix iLi z d x i d e d t
x e t
1 1 iz z e
branch cut
2
1
ln 11 ln
zt
Li d t i zt
On small circle, set
1 it e id t i e d
1t
On slanted line, set
1 i it r e ln 1 ln 1t t i
1r t
RHS of fig.18.8 & eq.18.159 are not allowed since the path crosses the branch cut.
Properties & Special Values 2
0
ln 1z
tLi z d t
t
2
1
n
n
z
n
2 0 0Li 2 21
11
n
Lin
2 2
2 16
Li
2 2
1
1n
n
Lin
2 1
1
n
sn
sn
1
1s
n
sn
2 ln 1d Li z z
d z z
2
2 2 1 ln ln 16
Li z Li z z z
2
22 2
1 1ln
6 2Li z Li z
z
22 2
1ln 1
1 2
zLi z Li z
z
2
2 112
Li
generates Li2 for all x from those in | x | 1/2.
Proof :both sides & find identity.Set z = 0 or 1 to determine const.
e.g.2
22
12 ln 2
2 6Li
Example 18.7.1. Check Usefulness of Formula
1 2 123 3
1 22 2 2 21 2 12
1
8
r r reI d r d r
r r r
12 1 2r r rj jr r
22
2 2
1 1ln
6 2Li Li
Question: Are the individual terms real?
I real & converges if , , 0 is real
Li2(x) is real for x < 1 both Li2 terms are real.
2ln
1 1
1 1
8. Elliptic Integrals
Example 18.8.1. Period of Simple Pendulum
2
21. .
2
dK E ml
d t
. . cosP E mgl
2
21cos
2
dE ml mgl
d t
cos Mmgl 0
M
d
d t
2cos cos M
d g
d t l
0
2 cos cos
M
M
l dt
g
2 2cos cos 2 sin sin2 2
MM
2 22sin cos2M
sin2sin
sin2M
cos1 2cos2 sin
2M
d d
2
cos cos cos2
M
dd
2 2
2
1 sin sin2M
d
2 20
4 4
1 sin sin2
M
M
l dT t
g
Period
Definitions
2 2
0
\1 sin sin
dF
Elliptic integral of the 1st kind
2 2
0
|1 1
x
d tF x m
t mt
0 1m 2
sin
sin
sin
t
x
m
/2
2
01 sin
dK m
m
Complete Elliptic integral of the 1st kind
1
2 20
1 |1 1
d tF m
t mt
2 2
0
\ 1 sin sinE d
Elliptic integral of the 2nd kind
2
2
0
1|
1
x
mtE x m dt
t
0 1m 2
sin
sin
sin
t
x
m
/2
2
0
1 sinE m d m
Complete Elliptic integral of the 1st kind
1
2
2
0
11 |
1
mtE m dt
t
Series Expansions /2
2
01 sin
dK m
m
/2
2
0
1 sinE m d m
/2
2
00
2 1 !!sin
2 !!n n
n
nK m d m
n
1/2
0
2 1 !!1
2 !!n
n
nx x
n
2
1
2 1 !!1
2 2 !!n
n
nm
n
Ex.13.3.8
Ex.18.8.2
2
1
2 1 !!1
2 2 !! 2 1
n
n
n mE m
n n
2 1
1 1, ;1 ;
2 2 2K m F m
2 1
1 1, ;1 ;
2 2 2E m F m
2 1
0
, ; ;!
nn n
n n
a b xF a b c x
c n
Fig.18.10. K(m) & E(m)
Mathematica
Limiting Values
2
1
2 1 !!1
2 2 !!n
n
nK m m
n
2
1
2 1 !!1
2 2 !! 2 1
n
n
n mE m
n n
0
2K
0
2E
1
2
2
0
1
1
mtE m dt
t
1
2 20
1 1
d tK m
t mt
1
2
0
11
d tK
t
1
0
1 1ln
2 1
t
t
1
0
1E dtIntegrals of the following form can be expressed in terms of elliptic integrals.
4
00
,
x
kk
k
I d t R t a t
E.Jahnke & F.Emde,
“Table of Higher Functions”
1