Complex Analysis: Theory and Applications · Problem 3.1.12

Analytic and Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · differentiation-rules-complex

Problem

Verify the following identities: (a)

\dfrac{d}{dz}C = 0

, where

C

is a constant: Identity (3.4). (b)

\dfrac{d}{dz}[f(z) + g(z)] = f'(z) + g'(z)

: Identity (3.7). (c)

\dfrac{d}{dz}\dfrac{f(z)}{g(z)} = \dfrac{g(z)f'(z)-f(z)g'(z)}{[g(z)]^2}

g(z) \neq 0

: Identity (3.9). (d)

\dfrac{d}{dz}f\!\left(g(z)\right) = f'\!\left(g(z)\right)g'(z)

: Identity (3.10). (e)

\dfrac{d}{dz}[f(z)]^n = n[f(z)]^{n-1}f'(z)

, where

n

is a positive integer: Identity (3.12).