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.

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- Oct 11th 2009, 11:02 AMbethhComplex 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.

- Oct 11th 2009, 12:46 PMTheEmptySet
The question you should be asking is when is $\displaystyle f(z)$ analytic then the formula $\displaystyle f'(z)=u_x+iv_x$ is valid. So check the cauchy-reimann equations

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

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

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

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