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Math Help - complex analysis question

  1. #1
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    complex analysis question

    one of the review questions that will help me study for my upcoming exam is as follows.

    Find all points z in C where the function h(z) = (z + 3)^2+i is differentiable. Find all points where this function is holomorphic. Assume that h is the principal value of the complex exponent.

    Any help would be appreciated since I am quite confused.
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  2. #2
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    Quote Originally Posted by shaheen7 View Post
    one of the review questions that will help me study for my upcoming exam is as follows.

    Find all points z in C where the function h(z) = (z + 3)^2+i is differentiable. Find all points where this function is holomorphic. Assume that h is the principal value of the complex exponent.

    Any help would be appreciated since I am quite confused.
    The question is either trivial or doesn't make any sense. (z+ 3)^2+ i is a polynomial and all polynomials are differentiable for all z. On the other hand, "holomorphic" means "analytic for all z" so it makes no sense to ask for points on which a function "is holomorphic".
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  3. #3
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    For the holomorphic part won't we need to use the Cauchy-Riemann equations? How would we go about finding du/dx, dv,dy and dv,dx, -dv,dy
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  4. #4
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by shaheen7 View Post
    one of the review questions that will help me study for my upcoming exam is as follows.

    Find all points z in C where the function h(z) = (z + 3)^2+i is differentiable. Find all points where this function is holomorphic. Assume that h is the principal value of the complex exponent.

    Any help would be appreciated since I am quite confused.
    For semplicity sake we set s=z+3 and neglect the constant term i so that the function becomes h(s)=s^{2}. Setting s=\sigma + i\ \omega You find that is...

    h(s) = u(\sigma, \omega) + i\ v(\sigma, \omega) = \sigma^{2}- \omega^{2} + 2\ i\ \sigma\ \omega

    ... and now it is easy to verify that the Cauchy Riemann equations are satisfied for any value of s...

    Kind regards

    \chi \sigma
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