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Math Help - Complex Variables differentiation

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    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.
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    Quote Originally Posted by bethh View Post
    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
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