Complex Analysis: Theory and Applications · Problem 3.2.16

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · wirtinger-derivative-cauchy-riemann

Problem

Recall that, for

z = x + iy

x = \dfrac{z+\bar{z}}{2}

and

y = \dfrac{z-\bar{z}}{2i}

. (a) Temporarily, think of

z

and

\bar{z}

as dummy symbols for real variables. With this perspective,

x

and

y

can be viewed as functions of

z

and

\bar{z}

. Use the chain rule for a function

h

of two variables to show that

\frac{\partial h}{\partial \bar{z}} = \frac{\partial h}{\partial x}\frac{\partial x}{\partial \bar{z}} + \frac{\partial h}{\partial y}\frac{\partial y}{\partial \bar{z}} = \frac{1}{2}\left(\frac{\partial h}{\partial x} + i\frac{\partial h}{\partial y}\right).

(b) Now define the operator

\dfrac{\partial}{\partial \bar{z}} = \dfrac{1}{2}\left(\dfrac{\partial}{\partial x} + i\dfrac{\partial}{\partial y}\right)

that is suggested by the previous equation. With this construct, show that if

f = u + iv

is differentiable at

z = (x,y)

, then, at the point

(x,y)

\dfrac{\partial f}{\partial \bar{z}} = \dfrac{1}{2}[u_x - v_y + i(v_x + u_y)] = 0

. Equating real and imaginary parts thus gives the complex form of the Cauchy-Riemann equations:

\dfrac{\partial f}{\partial \bar{z}} = 0