(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that:

[tex]\int_0^xJ_0(t)dt=2\sum_{n=0}^{\infty}J_{2n+1}(x)[/tex]

2. Relevant equations

I know that

[tex]J_0(t)=\sum_{s=0}^{\infty}\frac{(-1)^s}{s!s!}\frac{t^{2s}}{2^{2s}}[/tex]

3. 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 dont 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!

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# Homework Help: Bessel function problem

Can you offer guidance or do you also need help?

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