about bessel function integrals

[jayryu]

hello,everyone
i want to know how to solve this bessel function integrals:

\int_{0}^{R} J_m-1(ax)*J_m+1 (ax)*x dx
where J_m-1 and J_m+1 is the Bessel function of first kind, and a is a constant.

thanks.

[DuncanM]

Is this what you meant to post?

\int_{0}^{R} J_{m-1}(ax)*J_{m+1} (ax)*x \ dx

I am not an expert on Bessel functions, but isn't there an identity that you can use to simplify this expression? Something like

J_{m+1} = Some \ function \ of \ J_m

In other words, each successive Bessel function can be defined in terms of its predecessor. For example,

J1 = some function of J0,
J2 = some function of J1,
J3 = some function of J2,
J4 = some function of J3,
etc.

If you can find this identity, you should be able to simplify your integral.

[LCKurtz]

Perhaps you will find what you are looking for reading about the orthogonality of the Bessel functions. If you don't already know about that, try looking at:

http://www.hit.ac.il/ac/files/shk_b/Differential.Equations/Orthogonality_of_Bessel_functions.htm

[jayryu]

Perhaps you will find what you are looking for reading about the orthogonality of the Bessel functions. If you don't already know about that, try looking at:

http://www.hit.ac.il/ac/files/shk_b/Differential.Equations/Orthogonality_of_Bessel_functions.htm

thank you,i'll try that!

[jayryu]

Is this what you meant to post?

\int_{0}^{R} J_{m-1}(ax)*J_{m+1} (ax)*x \ dx

I am not an expert on Bessel functions, but isn't there an identity that you can use to simplify this expression? Something like

J_{m+1} = Some \ function \ of \ J_m

In other words, each successive Bessel function can be defined in terms of its predecessor. For example,

J1 = some function of J0,
J2 = some function of J1,
J3 = some function of J2,
J4 = some function of J3,
etc.

If you can find this identity, you should be able to simplify your integral.

yes,it is.thanks for your suggestions.i have solved it.:smile: