Complex Analysis: Theory and Applications · Problem 3.1.11

Analytic and Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · complex-differentiation

Problem

Show that the function f(z)=∣z∣2f(z) = |z|^2 is differentiable only at the point z0=0z_0 = 0. Hint: To show that ff is not differentiable at z0≠0z_0 \neq 0, choose horizontal and vertical lines through the point z0z_0 and show that ΔwΔz\dfrac{\Delta w}{\Delta z} approaches two distinct values as Δz→0\Delta z \to 0 along those two lines.

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