Show the convergence of a series

Question

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?

Answer 1

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.