A First Course in Complex Analysis · Problem 2.5.18

Differentiation — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · cauchy-riemann-equations

Problem

Prove Theorem 2.3.2: Suppose

f = u + iv

is a complex function defined in a neighborhood of

z_0 = x_0 + iy_0

, where

u

and

v

are real-valued. Then

f

is differentiable at

z_0

if and only if

u

and

v

are differentiable at

(x_0, y_0)

and satisfy the Cauchy--Riemann equations

u_x = v_y \quad \text{and} \quad u_y = -v_x

(x_0, y_0)

. Moreover,

f'(z_0) = u_x(x_0,y_0) + iv_x(x_0,y_0)