Complex Analysis: Theory and Applications · Problem 2.4.5

Analytic and Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · branches-cube-root

Problem

Let f1(z)=∣z∣13eiArg⁡(z)3=r13cos⁡θ3+ir13sin⁡θ3f_1(z) = |z|^{\frac{1}{3}}e^{i\frac{\operatorname{Arg}(z)}{3}} = r^{\frac{1}{3}}\cos\frac{\theta}{3} + ir^{\frac{1}{3}}\sin\frac{\theta}{3}, where ∣z∣=r≠0|z| = r \neq 0, and θ=Arg⁡(z)\theta = \operatorname{Arg}(z). f1f_1 denotes the principal cube root function. (a) Show that f1f_1 is a branch of the multivalued cube root f(z)=z13f(z) = z^{\frac{1}{3}}. (b) What is the range of f1f_1? (c) Where is f1f_1 continuous?

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