# Contour integration of the bessel function

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.
• $[x^k]\,f(x)$ is just a shorthand notation for "the coefficient of $x^k$ in the Taylor series of $f(x)$ around $x=0$". Dec 5 '14 at 9:49