# Bessel function problem

1. Jun 5, 2008

### b4b4

1. The problem statement, all variables and given/known data

Show that:
$$\int_0^xJ_0(t)dt=2\sum_{n=0}^{\infty}J_{2n+1}(x)$$

2. Relevant equations

I know that
$$J_0(t)=\sum_{s=0}^{\infty}\frac{(-1)^s}{s!s!}\frac{t^{2s}}{2^{2s}}$$

3. The attempt at a solution

I tried to calculate the integral and i get :
$$\sum_{s=0}^{\infty}\frac{(-1)^sx^{2s+1}}{s!s!2^{2s}(2s+1)}$$
I dont see how I can arrive to the reuslt, if I calclate :
$$\sum_{n=0}^{\infty}J_{2n+1}(x)=\sum_{s,n=0}^{\infty}\frac{(-1)^sx^{2s+2n+1}}{s!(s+2n+1)!2^{2n+2s+1}}$$
I tried to arrive to the same result, but I cant, shall I use Legendre's duplicantion formula?
But even if I use it I don't think that's the way
Thank you!

Last edited: Jun 5, 2008
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