Complex Analysis: Theory and Applications · Problem 9.3.10

Applications of Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · conformal-mapping-exercise

Problem

Find the image of the portion of the upper half-plane Im⁡(z)>0\operatorname{Im}(z) > 0 that lies outside the circle ∣z∣=1|z| = 1 under the transformation w=Log⁡ ⁣(1+z1−z)w = \operatorname{Log}\!\left(\frac{1+z}{1-z}\right).
Intuitive explanation

The key is recognizing that this transformation is built from simpler mappings you can compose: first a Möbius transformation that maps the exterior of the unit disk to a half-plane, then the logarithm that "unwraps" that half-plane into a vertical strip. Track where the boundary circles and rays go—the unit circle ∣z∣=1|z|=1 and the real axis will map to the edges of your image region, and the logarithm's branch cut determines which strip you land in.

Intuitive explanation

The key is recognizing that this transformation is built from simpler mappings you can compose: first a Möbius transformation that maps the exterior of the unit disk to a half-plane, then the logarithm that "unwraps" that half-plane into a vertical strip. Track where the boundary circles and rays go—the unit circle ∣z∣=1|z|=1 and the real axis will map to the edges of your image region, and the logarithm's branch cut determines which strip you land in.

Related theory

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