- #1
b4b4
- 1
- 0
Homework Statement
Show that:
[tex]\int_0^xJ_0(t)dt=2\sum_{n=0}^{\infty}J_{2n+1}(x)[/tex]
Homework Equations
I know that
[tex]J_0(t)=\sum_{s=0}^{\infty}\frac{(-1)^s}{s!s!}\frac{t^{2s}}{2^{2s}}[/tex]
The Attempt at a Solution
I tried to calculate the integral and i get :
[tex]\sum_{s=0}^{\infty}\frac{(-1)^sx^{2s+1}}{s!s!2^{2s}(2s+1)} [/tex]
I don't see how I can arrive to the reuslt, if I calclate :
[tex]\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}} [/tex]
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: