Complex Analysis: Theory and Applications · Problem 7.3.14

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · laurent-series

Problem

Show that

\cosh\!\left(z+\dfrac{1}{z}\right) = \displaystyle\sum_{n=-\infty}^{\infty} a_n z^n

, where the coefficients can be expressed in the form

a_n = \frac{1}{2\pi}\int_0^{2\pi}\cos n\theta\,\cosh(2\cos\theta)\,d\theta.

Hint: Let the path of integration be the circle

C_1^+(0)