Bessel 3

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17-May-13 MATH C241 prepared by MSR 1

Bessels Differentialequation

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In this lecture we discuss Bessels

Differential Equation. We also study

properties of Bessels functions, which are

solutions of Bessels equation. We first

review the definition and properties of the

celebrated Gamma function (which is also

called the extended factorial function).

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http://discovery.bits-

pilani.ac.in/discipline/math/msr/index.html

You may view my lecture slides in the

following site.

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Gamma Function (x)

This is easily one of the most importantfunctions in Mathematics.

Definition: For each real no x > 0, the

improper integral

dtte

xt 1

0

converges and its value is denoted by (x).

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(x + 1) = x(x) --- (Proved byIntegrating by parts)

(1) = 1 -- (immediate from the definition)

(n + 1) = n ! For all positive integers n

(follows from the first two properties; thusthe Gamma function is also referred to as

the extended factorial function)

Properties of the Gamma function (x)

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( ) (1 ) ( not an integer)sin

x x xx

We now extend the definition of(x) for

negative numbersx as follows:

If -1

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Thus we have extended the definition of(x)

for -1

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Proceeding like this, we define (x) for all

negative numbersx which are not negativeintegers. We also note that for allx (not a

negative integer),

( 1) ( )x x x

1 ( 1)x x x

And so on.

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We note that ifx is a positive integer,

(x + 1) =x(x) =x (x-1)(x1)

= ..

=x!We now define for all numbersx (not a

negative integer),

! ( 1)x x

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We also note for future reference that

! ( 1)( 2)..( )

( )!

x x x x n

x n

wherex is NOT a negative integer.

The next slide shows the graph of theGamma function.

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The second order homogeneous l.d.e.

2 2 2

( ) 0x y xy x p y wherep is anonnegative constant, is called as

the Bessels differential equation of orderp.

2

1( )

xP x

x x Here and

2 2 2

2 2( )( ) 1x p pQ x x x are analytic at all

points exceptx = 0.

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Hencex = 0 is the only singular point. As

( ) ( ) 1p x xP x 2 2 2

( ) ( )q x x Q x x p

andare both analytic atx = 0,x = 0 is a regular

singular point.

( 1) (0) (0) 0m m p m q

i.e. 2( 1) 0m m m p or2 2 0m p

Hence the exponents are m =

The indicial equation is

p, -p.

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Hence corresponding to the bigger exponent

m=p, there always exists a Frobenius Series

solution

00

( 0)p nnn

y x a x a

00

( 0)p nnn

a x a

1

0

( ) p nnn

y p n a x

2

0

( )( 1) p nnn

y p n p n a x

Hence

Substituting fory,y , y we get

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0

( )( 1) p nnn

p n p n a x

0

( ) p nnn

p n a x

2 2

0

( ) 0p n

n

n

x p a x

Divide throughout byxp, we get

2

0

( )( 1) ( ) nnn

p n p n p n p a x

2

0

0nnn

a x

In the second , replace n byn-2. We then get

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2 2

0

( ) nnn

p n p a x

22

0nnn

a x

Note that the constant term is absent. Thus

the above equation becomes

1 22

(2 1) (2 ) 0nn nn

p a x n p n a a x

Thus 1(2 1) 0p a

2(2 ) 0, 2n nn p n a a n

1 0a

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2

1, 2

(2 )n na a n

n p n or

Recurrence relation forans

n=3 gives 3 11

03(2 3)a ap

Putting n = 5, 7, we get

0 for all oddna n

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n = 2 gives2 0 02

1 1

2(2 2) 2 ( 1)a a a

p p

n = 4 gives4 2 22

1 1

4(2 4) 2 2( 2)a a a

p p

2

04

1( 1)

2 1 2 ( 1)( 2)a

p p

2

04

1( 1)

2 2! ( 1)( 2)a

p p

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Proceeding like this we get

2 02

1( 1)2 ! ( 1)( 2)...( )

n

n na a

n p p p n

n = 1, 2,

Hence the solution corresponding to the

exponent m =p is

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2

0 20

1( 1)

2 ! ( 1)( 2)...( )

p n n

nn

y a x xn p p p n

Now choosing 01

2 !pa

p we get the solution

2

0

1( 1)

! ( )! 2

n p

n

n

xy

n p n

This solution is denoted by ( )pJ x and is

referred to as Bessels function of the first kind.

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Thus2

0

1( ) ( 1)

!( )! 2

n p

np

n

xJ x

n n p

Ifp is a positive integer or zero, we see that

Jp(x) is a power series

Ifp is odd,Jp(x) contains only odd powersofx

Ifp is even,Jp

(x) contains only even powers

ofx

Jp(x) converges absolutely for allx > 0.

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Ifp is NOT an integer or zero, we easily

show that a second LI solution is2

0

1( ) ( 1)

!( )! 2

n p

n

pn

xJ x

n n p

Hence whenp is NOT an integer or zero,

the general solution of Bessels equation is

1 2( ) ( ),p py c J x c J x

c1, c2 arbitrary constants

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Ifp is a positive integerm, we formally define

2

0

1( ) ( 1)!( )! 2

n m

nm

n

xJ xn n m

Noting that1 1

0 for 0 ,( )! n mn m

we get2

1

( ) ( 1) !( )! 2

n m

n

mn m

x

J x n n m

Changing n to n+m, we get

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2

0

1( ) ( 1)

!( )! 2

n m

n mm

n

xJ x

n n m

2

0

1( 1) ( 1)

!( )! 2

n m

m n

n

x

n n m

( 1) ( )m

mJ x

HenceJm(x) andJ-m(x) are not LI.( m, a positive integer)

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Bessels Function of the second kind

We define the standard Bessel function

of the second kind by

( ) cos ( )( )

sin

p p

p

J x p J xY x

p

p not an integer.It is obvious that we can write the general

solution of Bessels equation of orderp also as

1 2( ) ( )p py c J x c Y x (c1, c2 arbitrary constants)

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It has been shown that ( ) lim ( )m pp m

Y x Y x

exists, is a solution of Bessels equation of

orderm and thatJm(x) and Ym(x) are LI.

Hence for allp (integer or not), the generalsolution of Bessels equation of orderp is

1 2( ) ( )p py c J x c Y x (c1, c2 arbitrary constants)

The following slides shows the graphs of

J0(x) ,J1(x), .. and Y0(x), Y1(x),

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