Complex Analysis: Theory and Applications · Problem 3.1.16

Analytic and Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · complex-differentiation

Problem

Let ff be differentiable at z0z_0. Show that there exists a function η(z)\eta(z) such that f(z)=f(z0)+f′(z0)(z−z0)+η(z)(z−z0),f(z) = f(z_0) + f'(z_0)(z - z_0) + \eta(z)(z - z_0), where η(z)→0\eta(z) \to 0 as z→z0z \to z_0.

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