Complex Analysis: Theory and Applications · Problem 2.5.1

Analytic and Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · reciprocal-transformation

Problem

In the following exercises find the image of the given circle or line under the reciprocal transformation

w = \frac{1}{z}

. (a) The horizontal line

\operatorname{Im}(z) = \frac{1}{5}

. (b) The circle

C_{\frac{1}{2}}\!\left(-\frac{i}{2}\right) = \left\{z : \left|z + \frac{i}{2}\right| = \frac{1}{2}\right\}

. (c) The vertical line

\operatorname{Re}(z) = -3

. (d) The circle

C_1(-2) = \{z : |z + 2| = 1\}

. (e) The line

2x + 2y = 1

. (f) The circle

C_1\!\left(\frac{i}{2}\right) = \left\{z : \left|z - \frac{i}{2}\right| = 1\right\}

. (g) The circle

C_1\!\left(\frac{3}{2}\right) = \left\{z : \left|z - \frac{3}{2}\right| = 1\right\}

. (h) The circle

C_2(-1+i) = \{z : |z + 1 - i| = 2\}