Conformal mappings

Objective. Transform domains while preserving local angles.

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Learn: the key idea

A conformal mapping is a holomorphic function with nonzero derivative that preserves angles (both magnitude and orientation) between curves at each point where it is applied. Because f'(z₀)≠0, locally f acts like multiplication by the complex number f'(z₀), which rotates and scales but never distorts angles. Möbius transformations f(z)=(az+b)/(cz+d) with ad−bc≠0 are conformal everywhere except possibly at poles, and they map circles and lines to circles or lines. For example, f(z)=1/z maps the circle |z|=1 to itself and sends the upper half-plane to the lower half-plane while preserving angles throughout.

Worked example

Determine where the conformal mapping f(z)=z³+1 fails to be angle-preserving (locate its critical points).

1 $Conformality fails where f'(z)=0$
2 $Compute f'(z)=3z²$
3 $Set 3z²=0, giving z=0$
4 $Critical point is z=0$

Answer: z = 0

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Common mistake. Students assume a conformal map preserves angles even at critical points where f'(z₀)=0; at such points the angle-preserving property fails and angles get multiplied by an integer factor instead, as happens for f(z)=z² at z=0, which doubles angles.

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