Complex Analysis: Theory and Applications · Problem 4.3.12

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · geometric-series-real-imaginary-parts

Problem

This exercise makes interesting use of the geometric series. ((a)) Use the formula for geometric series with

z = re^{i\theta}

, where

r < 1

, to show that

\sum_{n=0}^{\infty} z^n = \sum_{n=0}^{\infty} r^n e^{in\theta} = \frac{1 - r\cos\theta + ir\sin\theta}{1 + r^2 - 2r\cos\theta}.

((b)) Use part (a) to obtain

\sum_{n=0}^{\infty} r^n \cos n\theta = \frac{1 - r\cos\theta}{1 + r^2 - 2r\cos\theta}, \quad \text{and}

\sum_{n=0}^{\infty} r^n \sin n\theta = \frac{r\sin\theta}{1 + r^2 - 2r\cos\theta}.