A First Course in Complex Analysis · Problem 22

Isolated Singularities and the Residue Theorem — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · power-series

Problem

(a) Find the power series representation for exp⁡(az)\exp(az) centered at 00, where a∈Ca \in \mathbb{C} is any constant. (b) Show that exp⁡(z)cos⁡(z)=12(exp⁡((1+i)z)+exp⁡((1−i)z)).\exp(z)\cos(z) = \frac{1}{2}\left(\exp((1+i)z) + \exp((1-i)z)\right). (c) Find the power series expansion for exp⁡(z)cos⁡(z)\exp(z)\cos(z) centered at 00.

Related theory

Browse the problem library