Proving Bessel Integral Relation

[rizardon]

Homework Statement

Show that the integral (from 0 to infinity) of e^-axJ_p(bx)dx = [sqr(a²+b²) - a] / b^psqr(a²+b²)

Homework Equations

J_p(bx) =
summation [ (-1)^k(bx/2)^2k+p ] / k! Gamma(k+p+1)

Gamma(x) = integral (from 0 to infinity) of e^-tt^x-1dt

The Attempt at a Solution

I've changed the J_p(bx) to its series representation

integral (from 0 to infinity) of e^-ax * summation [ (-1)^k(bx/2)^2k+p ] / k! Gamma(k+p+1) dx

Next I represent the integral part with the gamma function

summation [ (-1)^k (b)^2k+p Gamma (2k+p+1) ] /
(k!)(2)^2k+p Gamma(k+p+1) (a)^2k+p+1


I'm following the steps given by the book (Special Functions of Mathematics For Engineers by Larry C. Andrews p.256). The pattern in the book is to apply the Legendre duplication formula and then try to manipulate the expression to express the expression as a binomial series. Somehow I don't think the duplication formula will work here. Can anyone tell me what I should do for the next step or is there a better approach to prove the equality.

 

[lanedance]

haven't tried it myself, but have you tried integrating by parts and using some of the derivatives identities for bessel functions?

 

[lanedance]

this site has some useful properties
http://www.efunda.com/math/bessel/bessel.cfm

 

[rizardon]

The identities are all for J_p(x). Can I just replace x with bx or do i need to apply other properties to do so. Thanks.

 

[lanedance]

i''d test to see whether it works for b=1 before complicating things but the same formulas can be found on wiki
http://en.wikipedia.org/wiki/Bessel_function#Derivatives_of_J.2CY.2CI.2CH.2CK