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Given the series:

$S(\lambda,\Phi)=\sum_{n=0}^{\infty}{J_n}(\lambda)e^{in\Phi}$

where the $J_n(\lambda)$ is the Bessel function of order $n$

I have some difficulty to give a proof of its convergence. Can someone help me?

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This series is well-known in the theory of the Bessel functions and Fourier series being the representation of

$S(\lambda,\Phi)=\sum_{n=0}^\infty J_n(\lambda)e^{in\Phi}=e^{i\lambda\sin\Phi}$

and this Fourier series is converging as can be seen from the fact that

$\left|\sum_{n=0}^\infty J_n(\lambda)e^{in\Phi}\right|<\sum_{n=0}^\infty |J_n(\lambda)|<\infty.$

for $\lambda$ finite.

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    @NickThompson: Yours is a good question but I know the sum of the series that is defined for any finite $\lambda$. You can also wok this out through the coefficients of the Fourier series $c_n=\frac{1}{2\pi}\int_0^{2\pi}e^{i\lambda\sin\Phi-in\Phi}d\Phi$.2012-02-14