Complex Analysis: Theory and Applications · Problem 9.1.2

Conformal Mapping — Howell & Matthews

Mathematics — Complex Analysis · conformal-mappings-angle-rotation-scale

Problem

Find the angle of rotation

\alpha = \operatorname{Arg}(f'(z))

and the scale factor

|f'(z)|

of the mapping

w = f(z)

at the indicated points. (a)

w = \dfrac{1}{z}

at the points

1

1+i

, and

i

. (b)

w = \ln r + i\theta

, where

-\pi < \theta \leq \pi

, at the points

1

1+i

i

, and

-1

. (c)

w = r^{\frac{1}{2}}\cos\dfrac{\theta}{2} + ir^{\frac{1}{2}}\sin\dfrac{\theta}{2}

, where

-\pi < \theta < \pi

, at the points

i

1

-i

, and

3+4i

. (d)

w = \sin z

at the points

\dfrac{\pi}{2}+i

0

, and

-\dfrac{\pi}{2}+i