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Math Help - Analyticity of a function of complex variable

  1. #1
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    Analyticity of a function of complex variable

    Hi! I'm not quite sure if this belongs in this subforum due to the complex variable, but I couldn't find any other logical place to ask this question.

    Here's the problem I have troubles solving:

    Given f(z)=|z|sin(z) , determine f'(z), where it exists, and state where f is analytic and where it is not.

    Of course, z is a complex variable such as z=x+iy.

    I've first tried with the Cauchy Riemann equations, but I end up with something quite messy that is unsolvable from what I see.

    How should I proceed?

    Thanks.
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  2. #2
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    Re: Analyticity of a function of complex variable

    I started trying something and I get a messy equation like you said, so someone else could fill it on this. But, im quite certain that a function like this is at best differentiable at only a few places; perhaps only a set of isolated points, and certainly not any region. This is because |z| is quite un-analytic and it messes with the structure of the function. Hope this helps.
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  3. #3
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    Re: Analyticity of a function of complex variable

    Thanks from topsquark
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  4. #4
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    Re: Analyticity of a function of complex variable

    A product of functions is holomorphic everywhere that both of the factors are holomorphic. sin(z) is holomorphic everywhere. Where is |z| holomorphic?
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  5. #5
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    Re: Analyticity of a function of complex variable

    So I've found that the reason the equation I ended up with seemed long and hard to find the solutions was because I didn't think of distributing one term from each side of the equation, which would end up simplifying it to what tom gas posted. I thought of it after seeing the simplified equation and noticing similar terms from mine.

    After this I just isolated x/y from each equation and equalized them, finding that the equation was only differentiable at (n*pi, 0) for n=+-1,+-2,+-3 (not 0 because we have a fraction with x and y at the bottom) and so on, thus that it was analytic nowhere.

    Can't believe it took me 5 hours trying to solve this exercise because I forgot to distribute. \o/

    Thanks for your help!
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