Complex Analysis: Theory and Applications · Problem 7.3.16

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · laurent-series-bessel

Problem

The Bessel function

J_n(z)

is sometimes defined by the generating function

\exp\!\left[\frac{z}{2}\!\left(t-\frac{1}{t}\right)\right] = \sum_{n=-\infty}^{\infty} J_n(z)t^n.

Use the circle

C_1^+(0)

as the contour of integration and show that

J_n(z) = \frac{1}{\pi}\int_0^{\pi}\cos(n\theta - z\sin\theta)\,d\theta.