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Math Help - Complex anaylsis

  1. #1
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    Complex anaylsis

    Use the definition to show that f'(z) does not exist when f(z)=Re(z).

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  2. #2
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    Quote Originally Posted by LCopper2010 View Post
    Use the definition to show that f'(z) does not exist when f(z)=Re(z).

    Thanks!

    In case the limit exists we have f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}=\lim_{z\to z_0}\frac{Re(z)-Re(z_0)}{z-z_0} , so put z=x+yi\,,\,\,z_0=a+bi:

    f'(z_0)=\lim_{z\to z_0}\frac{x-a}{(x-a)+(y-b)i} . Now, choose y=b and make x\rightarrow a , so:

    \lim_{x\to a}\frac{x-a}{x-a}=1 , and now choose x=a and make y\rightarrow b

    \lim_{y\to b}\frac{0}{(y-b)i}=0.

    Since the wanted limit depends on how the complex variable z approaches a+bi the limit, and thus the derivative, doesn't exist.

    Tonio
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