Integration of Bessel function

[johnsmi]

Homework Statement

Hi, I need to integrate this:
\int(J₀(r))²rdr between 0<r<a
It is for calculating the energy of a nondiffracting beam inside a radius of a. (the r is because of the jacobian in polar coordinates)

The Attempt at a Solution

I saw somewhere that said the integral was a known one and it didn't even bother to write it down but couldn't find it anywhere else
I think it might be found also in Ballanis:antenna theory but I can't find the book

Thanks in advance

 

[phyzguy]

Mathematica says that:
\int^a_0J_0(r)^2 rdr = \frac{a^2}{2}(J_0(a)^2+J_1(a)^2)