Exploring the Relationship Between g(x,t) and J_n(x)

[Another1]

\(\displaystyle g(x,t) = e^{(\frac{x}{2})(t-\frac{1}{t})}=\sum_{n=-\infty}^{\infty}J_{n}(x)t^{n}\)

and

\(\displaystyle \left| J_{0}(x) \right|\le 1 \) and \(\displaystyle \left| J_{n}(x) \right|\le \frac{1}{\sqrt{2}} \)

how to show that

1=\(\displaystyle (J_{0}(x))^{2}+2(J_{1}(x))^{2}+2(J_{2}(x))^{2}+...\)

I don't have idea

 

[topsquark]

Re: [please help]Bessel function

\(\displaystyle g(x,t) = e^{(\frac{x}{2})(t-\frac{1}{t})}=\sum_{n=-\infty}^{\infty}J_{n}(x)t^{n}\)

and

\(\displaystyle \left| J_{0}(x) \right|\le 1 \) and \(\displaystyle \left| J_{n}(x) \right|\le \frac{1}{\sqrt{2}} \)

how to show that

1=\(\displaystyle (J_{0}(x))^{2}+2(J_{1}(x))^{2}+2(J_{2}(x))^{2}+...\)

I don't have idea

I haven't done the whole thing but note that Bessel functions are orthogonal:
\(\displaystyle \int _0 ^1 J_{\nu} \left ( \alpha_{\nu ~ m} x \right ) ~ J_{\nu } \left ( \alpha_{\nu ~ n} x \right ) ~x ~dx = \frac{1}{2} \left [ J_{\nu + 1} \left ( \alpha _{\nu ~ m} \right ) \right ] ^2 \delta_ {m~n}\)

Does this give you any ideas?

-Dan