The Bessel Function $J_n(x)$ is defined, for a natural number $n$ and real number x, as
$J_n(x) = \frac{1}{2\pi}\int_0^{2\pi}\cos(n\theta-x\sin\theta)d\theta.$
By using contour integration with integrand $z^{n-1}\exp(\frac{-xz}{2})\exp(\frac{x}{2z})$, or otherwise, show that
$J_n(x)=\sum_{\substack{k=0}}^\infty\frac{(-1)^k}{k!(n+k)!}(\frac{x}{2})^{n+2k}$
Hi, can someone give me some hints or give a simple example to convert the integral to a sum form using integrand. I am not very familiar with how to use integrand.
Many many thanks!
From the integral definition we have: $$J_n(x) = \frac{1}{2\pi}\Re\int_{0}^{2\pi}e^{-ix\sin\theta}e^{ni\theta}\,d\theta $$ and expanding the exponential function as a Taylor series we get: $$ [x^k]\,J_n(x) = \frac{1}{2\pi}\Re\int_{0}^{2\pi}\frac{(-i\sin\theta)^k}{k!}e^{ni\theta}\,d\theta, $$ so, by expressing $\sin\theta$ as $\frac{e^{i\theta}-e^{-i\theta}}{2i}$, using the binomial theorem and the identity: $$ \frac{1}{2\pi}\int_{0}^{2\pi}e^{ni\theta}e^{-ki\theta}\,d\theta = \delta_{n,k}$$ we easily prove our claim.