Annoying Integral involving Bessel Functions

**thelostchild** · February 20th 2011, 11:18 AM

Hi I'm having trouble with this integral
$ \int_{0}^{\infty} \frac{J_0(kR)}{(1+(kR_d)^2)^{3/2}} dk $

I'm supposed to evaulate it using
$ \int_{0}^{\infty} J_{\nu}(xy) \frac{dx}{(x^2+a^2)^{1/2}} = I_{\nu/2} (ay/2) K_{\nu/2} (ay/2) $

Where standard notation has been used for the bessel functions, any hints on how to transform it to the correct forn would be much appreciated, I can't really see how to get this to work

**topsquark** · February 20th 2011, 12:42 PM

Originally Posted by thelostchild

Hi I'm having trouble with this integral
$ \int_{0}^{\infty} \frac{J_0(kR)}{(1+(kR_d)^2)^{3/2}} dk $

I'm supposed to evaulate it using
$ \int_{0}^{\infty} J_{\nu}(xy) \frac{dx}{(x^2+a^2)^{1/2}} = I_{\nu/2} (ay/2) K_{\nu/2} (ay/2) $

Where standard notation has been used for the bessel functions, any hints on how to transform it to the correct forn would be much appreciated, I can't really see how to get this to work

Hint:
$\displaystyle \int_0^{\infty} \frac{J_0(kR)}{(1 + (kR)^2 )^{3/2}}dk = ~...~= \frac{1}{y}\int_0^{\infty}J_0(m) \cdot m(1 + m^2)^{-3/2}dm$
(after letting y = R and m = xy.)

Now integrate by parts:
$\displaystyle \int p~dq = pq - \int q ~dp$
using $\displaystyle p = J_0(m)$ and $\displaystyle dq = m(1 + m^2)^{-3/2}$

-Dan

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