Complex Analysis: Theory and Applications · Problem 3.3.15

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · logarithm-equipotentials-streamlines

Problem

We introduce the logarithmic function in Chapter 5. For now, let F(z)=Log⁡z=ln⁡∣z∣+iArg⁡zF(z) = \operatorname{Log} z = \ln|z| + i\operatorname{Arg} z. Here we have ϕ(x,y)=ln⁡∣z∣\phi(x,y) = \ln|z| and ψ(x,y)=Arg⁡z\psi(x,y) = \operatorname{Arg} z. Sketch the equipotentials ϕ=0,ln⁡2,ln⁡3,ln⁡4\phi = 0, \ln 2, \ln 3, \ln 4 and streamlines ψ=kπ8\psi = \dfrac{k\pi}{8} for k=0,1,…,7k = 0, 1, \ldots, 7.

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