A First Course in Complex Analysis · Problem 43

Integration — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · holomorphic-functions

Problem

For each of the following functions, determine all complex numbers for which the function is holomorphic. If you run into a logarithm, use the principal value unless otherwise stated. ((a)) zˉ2\bar{z}^2 Answer. differentiable at 0, nowhere holomorphic ((b)) sin⁡zz3+1\frac{\sin z}{z^3+1} Answer. differentiable and holomorphic on C∖{−1, eiπ3, e−iπ3}\mathbb{C} \setminus \{-1,\, e^{i\frac{\pi}{3}},\, e^{-i\frac{\pi}{3}}\} ((c)) L ⁣og⁡(z−2i+1)\mathcal{L}\!\operatorname{og}(z-2i+1) where L ⁣og⁡(z)=ln⁡∣z∣+iArg⁡(z)\mathcal{L}\!\operatorname{og}(z) = \ln|z| + i\operatorname{Arg}(z) with 0≤Arg⁡(z)<2π0 \leq \operatorname{Arg}(z) < 2\pi Answer. differentiable and holomorphic on C∖{x+iy∈C:x≥−1, y=2}\mathbb{C} \setminus \{x+iy \in \mathbb{C} : x \geq -1,\, y = 2\} ((d)) exp⁡(zˉ)\exp(\bar{z}) Answer. nowhere differentiable or holomorphic ((e)) (z−3)i(z-3)^i Answer. differentiable and holomorphic on C∖{x+iy∈C:x≤3, y=0}\mathbb{C} \setminus \{x+iy \in \mathbb{C} : x \leq 3,\, y = 0\} ((f)) iz−3i^{z-3} Answer. differentiable and holomorphic in C\mathbb{C} (i.e., entire)

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