[Math] Laurent series and Bessel functions

Question

I'm working on a problem in complex analysis that I don't know how to approach. The problem is as follows:

Let the Bessel functions $J_n$, for integer $n$, be defined by

$$e^\left(\frac{w}{2}(z-\frac{1}{z}\right)=\sum_{n=-\infty}^{+\infty}J_{n}(w)z^n. $$

Show that

$$J_{n} = \frac{1}{2\pi}\int_{0}^{2\pi}\exp(iw\sin\theta-in\theta)d\theta. $$

Any pointers welcome – thanks!

Answer 1

Hint: The right hand side of the defining equation is a Laurent series. How do you compute the coefficients of a Laurent series for a given function (e.g., the left hand side of the defining equation) via contour integrals?