Math Help - Complex Variables differentiation

1. Complex Variables differentiation

Show that when f(z)=x^3+i(1-y)^3, it is legitimate to write f'(z)=ux+ivx=3x^2 only when z=i.

2. Originally Posted by bethh
Show that when f(z)=x^3+i(1-y)^3, it is legitimate to write f'(z)=ux+ivx=3x^2 only when z=i.
The question you should be asking is when is $f(z)$ analytic then the formula $f'(z)=u_x+iv_x$ is valid. So check the cauchy-reimann equations

$u(x,y)=x^3;v(x,y)=(1-y)^3$

$u_x=3x^2$ and $v_y=-3(1-y)^2$

Now for what points (x,y) does $u_x=v_y$

this gives $3x^2=-3(1-y)^2 \iff 3(x^2+(1-y)^2)=0$ The only solution to this equation is (0,1) this is the same as the complex number $z=i$ also note that $u_y=-v_x=0$