Cauchy-Riemann Equations and Complex Derivatives: A Homework Problem

[Benzoate]

Homework Statement

Show that when f(z)=x^3+i(1-y)^3, it is legitimate to write:

f'(z)=u_x+iv_x=3x^2
only when z=i

Homework Equations

Cauchy riemann equations:

u_x=v_y , u_y=-v_x
f'(z)=u_x+i*v_y

The Attempt at a Solution

u=x^3
v=(1-y)^3
u_x=3*x^2
v_y=-3*(1-y)^2
x^2=-(1-y)^2 =

u_y=0
-v_x=0

f'(z)=3*x^2+i(0)= 3*x^2

I don't understand why z=i => z=o+i*1? is relevant to show that f'(z)=3*x^2

 

[Dick]

f'(z) only has a complex derivative if it satisfies the Cauchy-Riemann equations. You have correctly found that means x^2=-(1-y)^2. How many values of x and y satisfy that?