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Thread: Complex differentiable and holomorphic functions

  1. #1
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    Complex differentiable and 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 is 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.


    h is differentiable only at $\displaystyle z_{0}=0$

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

    f and h are nowhere holomorphic.

    if my answer for the first part is right, then g is holomorphic in $\displaystyle (-\infty , 0)$ and $\displaystyle (0, \infty)$. But now that I think of it it doesn't really make much sense.

    Can someone please check these answers for me?
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  2. #2
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    Notice that g is not really a polynomial in $\displaystyle \mathbb{C}$. As for the other question, a function f is holomorphic at z if it's (complex) differentiable in an open ball around z.
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  3. #3
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    Quote Originally Posted by Jose27 View Post
    Notice that g is not really a polynomial in $\displaystyle \mathbb{C}$.
    I'm not sure I see why g is not a polynomial in $\displaystyle \mathbb{C}$.

    As for the other question, a function f is holomorphic at z if it's (complex) differentiable in an open ball around z.
    I know the definition of holomorphic but I'm not sure if I understand how to apply it.

    Do my answers to both questions seem right to you? What about the differentiability of g?
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  4. #4
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    Notice that if then is not a lower bound for .
    So .
    In this way construct a decreasing sequence of distinct terms from converging to .

    Consider this set of nonnegative numbers: .

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