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Math Help - Show a complex function is differentiable at 0

  1. #1
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    Show a complex function is differentiable at 0

    (Sorry for all the questions on this, I am really struggling with this and really appreciate all help.)

    Show that the function f(z)=|z|^2 is differentiable at the origin.


    So if z=x+iy, then |z|^2=x^2+y^2.

    To show that it's differentiable you have to show that the limit of the difference quotient exists at 0:

    \lim_{\Delta z\to 0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}

    =\lim_{\Delta z\to0}\frac{f(0+\Delta z)-f(0)}{\Delta z}

    =\lim_{\Delta z\to0}\frac{(\Delta x)^2+(\Delta y)^2-0}{\Delta x+i\Delta y}

    Is this the right way to start?
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  2. #2
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    Re: Show a complex function is differentiable at 0

    Hey Ragnarok.

    Hint: You might want to try looking at the Cauchy-Riemann equations. Have you come across these in class?
    Thanks from Ragnarok
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  3. #3
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    Re: Show a complex function is differentiable at 0

    Ah, we have covered the Cauchy-Riemann equations, though technically this problem is from a section before they are covered. I think it will work fine, though, thanks for the hint!
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