Coefficients of a Fourier-Bessel series

[ebernardes]

When finding the coefficients of a Fourier-Bessel series, the Bessel functions satisfies, for k_1and k_2 both zeroes of J_n(t), the orthogonality relation given by:
$$\int_0^1 J_n(k_1r)J_n(k_2r)rdr = 0, (k_1≠k_2)$$
and for k_1 = k_2 = k:

$$\int_0^1 J_n^2(kr)rdr = \frac12J_n^{'2}(k)$$

I understand how to get the first result since the Bessel's equation can be interpreted as a Sturm-Liouville problem, but how can I show the second one?

 

[SteamKing]

This paper might offer a clue:

 

[ebernardes]

This paper might offer a clue:

But how exactly i can evaluate the integral and get the result given by: I = \frac{R^2}{\alpha_m^2 - \alpha_n^2}[\alpha_mJ_0(\alpha_n)J_1(\alpha_m) - \alpha_nJ_0(\alpha_m)J_1(\alpha_n)] or another more general formula for Bessel functions of different order?

 

[SteamKing]

I believe an integration by parts is called for, using certain relations of Bessel functions to get over the tricky bits:

http://home.comcast.net/~rmorelli146/U3150/Bessel.pdf