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Thread: Holomorphic functions

  1. #1
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    Holomorphic functions

    (a) At what points $\displaystyle z \in \mathbb{C}$ are the functions $\displaystyle f(z)=|z|^4$ and $\displaystyle g(z)=g(x+iy)=6xy^{2}+i(4x+2y^{3})$ and $\displaystyle h(z)=z \overline{z}^{2}$ differentiable? At what points are f and g and h holomorphic?

    Using Cauchy-Riemann equations (+ showing continuity of partial derivatives) I have found:

    f and h are differentiable only at $\displaystyle z_{0}=0$

    For g, solving for x and y in Cauchy-Riemann I end up with $\displaystyle 6y^{2}=6y^{2}$ and $\displaystyle xy=-\frac{1}{3}$ so $\displaystyle z_{0}$ is in the form $\displaystyle z_{0}=x-\frac{1}{3x}i$. Now, $\displaystyle \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}=\frac{2}{3x^{2}}$ are continuous $\displaystyle \forall x \in \mathbb{R}$\{$\displaystyle 0$} and since $\displaystyle x=\frac{-1}{3y}$ we must have $\displaystyle x,y \in \mathbb{R} $\{$\displaystyle 0$}. So, g should be differentiable $\displaystyle \forall z_{0} \in \mathbb{C}$\{$\displaystyle 0$} satisfying $\displaystyle x=\frac{-1}{3y}$.
    But I have read that polynomial with coefficients in $\displaystyle \mathbb{C}$ are differentiable in $\displaystyle \mathbb{C}$. Hence, I'm not sure about my answer for g.


    As to where the functions are holomorphic, I'm not quite sure I understand the concept very well. This is what I have found:

    f and h are nowhere holomorphic.

    I am not sure how to approach this for the function g.

    Can someone please check these answers for me?
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  2. #2
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    Holomorphic means differentiable in a neighborhood of a point. The functions (a) and (c) are not differentiable in a "region" about a point.

    As for (b) don't confuse a "complex polynomial" with a "real polynomial" with complex coefficients. Now, the complex polynomial $\displaystyle f(z)=a_0 z^n+a_1 z^{n-1}+\cdots+a_n$ is entire but your function is not of that form but rather just two real functions with an "i" coefficient in front of the second one.

    And the function $\displaystyle f(x,y)=g(x,y)+ih(x,y)$ is complex analytic iff h(x,y) is the complex congugate of g(x,y).
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