Bessel integration by parts

• Feb 26th 2009, 02:57 PM
canopy
Bessel integration by parts
Hi guys. I'm totally stuck on this problem. Any help would be appreciated:

Using the definitions for the derivatives of Jp(x) (Bessel Function), along with integration by parts, demonstrate the following...

http://imgfreehost.com/out.php?i26646_Picture4.png

Edit: Embedding of image isn't working for some reason. Here's a link...

http://imgfreehost.com/out.php?i26646_Picture4.png
• Feb 26th 2009, 03:56 PM
Jester
Quote:

Originally Posted by canopy
Hi guys. I'm totally stuck on this problem. Any help would be appreciated:

Using the definitions for the derivatives of Jp(x) (Bessel Function), along with integration by parts, demonstrate the following...

http://imgfreehost.com/out.php?i26646_Picture4.png

Edit: Embedding of image isn't working for some reason. Here's a link...

http://imgfreehost.com/out.php?i26646_Picture4.png

I'll do the first, the second follows similarly. We have the following property

$xJ_n' = - n J_n + x J_{n-1}$

so

$xJ_{n+1}' = - (n+1) J_{n+1} + x J_{n}$

then

$x^mJ_{n+1}' = - (n+1) x^{m-1}J_{n+1} + x ^mJ_{n}$

so

$\int x^mJ_{n+1}' dx = - (n+1)\int x^{m-1}J_{n+1} dx+ \int x ^mJ_{n}dx$

integration by parts on the frist integral gives

$x^m J_{n+1} - m \int x^{m-1} J_{n+1}' dx = - (n+1)\int x^{m-1}J_{n+1} dx+ \int x ^mJ_{n}dx$

and re-arranging terms gives

$\int x ^mJ_{n}dx = x^m J_{n+1} - ( m - n-1)\int x^{m-1}J_{n+1} dx$

For the second, try using

$xJ_n' = n J_{n} - xJ_{n+1}$
• Feb 26th 2009, 04:34 PM
canopy
Thanks!
In the interest of good taste, I'll refrain from using multiple exclamation points and all capital letters, but it's very hard to resist.

Thanks!