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Math Help - Holomorphic functions

  1. #1
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    Oct 2008
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    Holomorphic functions

    (a) At what points z \in \mathbb{C} are the functions f(z)=|z|^4 and g(z)=g(x+iy)=6xy^{2}+i(4x+2y^{3}) and 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 z_{0}=0

    For g, solving for x and y in Cauchy-Riemann I end up with 6y^{2}=6y^{2} and xy=-\frac{1}{3} so z_{0} is in the form z_{0}=x-\frac{1}{3x}i. Now, \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}=\frac{2}{3x^{2}} are continuous \forall x \in \mathbb{R}\{ 0} and since x=\frac{-1}{3y} we must have x,y \in \mathbb{R} \{ 0}. So, g should be differentiable \forall z_{0} \in \mathbb{C}\{ 0} satisfying x=\frac{-1}{3y}.
    But I have read that polynomial with coefficients in \mathbb{C} are differentiable in \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
    Super Member
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    Aug 2008
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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 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 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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