Exercise [08.04]

Sameed Zahoor · 13 Jul 2008, 14:36

The equation of a circle in z-plane is given by

$z\overline{z}=k$ where k is a constant.

For taking any curve into w-plane we need to take every z in z-plane into w-plane using the given transformation.Thus,

$z\rightarrow \frac{1}{z}$ and

$\overline{z}\rightarrow\frac{1}{\overline{z}}$

The given circle in the z-plane can also be represented as

$\frac{1}{z\overline{z}}=\frac{1}{k}$ (z is not zero)

In the w-plane this amounts to,

$w\overline{w}=\frac{1}{k}$

which is again the equation of a circle.Hence,actual circles are mapped to actual circles by the reciprocal function.

Exercise [08.04]

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