How Do You Solve 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.