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Math Help - Limit / Differentiability of complex functions

  1. #1
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    Limit / Differentiability of complex functions

    I am new to delta-epsilon arguments to prove the existence of limits.

    Q - Find all the points on the complex plain where f(z) = |z|^2 is differentiable? i.e. I seek all z0 where lim z->z0 [f(z)-f(z0)]/[z-z0] exisits

    Can someone please help me with a nice delta-epsilon argument to solve this?

    I did it without using any delta-epsilon argument. Answer I got is only z0 = 0 to be the only point.

    Would really appreciate is someone can help me with a rigorous argument based on delta-epsilon definition of existence of limit.
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  2. #2
    MHF Contributor chisigma's Avatar
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    If we write z= x + i y and f(z)= u(x,y) + iv(x,y) , then f(z) is differentiable [analitic] where are satisfied the Cauchy-Riemann conditions...

    \frac{du}{dx} = \frac{dv}{dy}

    \frac{du}{dy} = -\frac{dv}{dx} (1)

    In this case is...

    u(x,y) = x^{2} + y^{2}

    v(x,y)=0 (2)

    ... and the (1) are satisfied only in z=0...

    Kind regards

    \chi \sigma
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  3. #3
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    Thanks. I do not want to use CR equations just want to use an epsilon-delta argument. I want to do this just to understand/practice how to use the most basic defintions of continity ans differentiability to prove this without any other tool at hand.
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