Jacobi series for Bessel's function

Question

Using Jacobi series, prove the following

${J_0}^2+{J_1}^2+{J_2}^2+\cdots=1$

My trial:

$\cos(x\sin θ)=J_0+2J_2\cos(2θ) +2J_4\cos(4θ)+\cdots$

$\sin(x\sinθ)=2\big(J_1\sin(θ) +J_3\sin(3θ)+J_5\sin(5θ)+\cdots\big)$

Squaring both equations and adding them is what I thought we are supposed to do but in this is not enough to solve this equality. How do i proceed from here

The ()'s are oddly placed, please fix that. – emacs drives me nuts Feb 14 '20 at 08:24 — emacs drives me nuts, Feb 14 '20 at 08:24
Weird equations, there is no $x$ on the RHS. – vonbrand Feb 14 '20 at 13:21 — vonbrand, Feb 14 '20 at 13:21

score 0 · Answer 1 · edited Oct 18 '22 at 06:08

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Let both of the Jaccobi series then square both of the equations. Now, first take the $\cos^2$ series and apply integration with limits $0$ to $\pi$. which then gives a result of an even number, and then take the second series and repeat

edited Oct 18 '22 at 06:08

answered Oct 18 '22 at 05:22

haris

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Jacobi series for Bessel's function

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