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Show this is analytic

  1. Dec 9, 2008 #1
    This isn't my whole question, just part of the question I am trying to do to show the whole thing is analytic.

    I can do the rest but showing this is analytic:


    Is trickey for me..

    I am trying to show it is analytic by showing it satisfies the cauchy riemann equations.

    I take z = x + iy

    And my function turns into (after simplifying)

    [x^3 - 3xy^2 + 1 + i(3yx^2 - y^3)] / (x + iy -1)

    Now I can split the numerator in real and imaginary parts, but the denominator has an i in it which is in the way for me, hence I can't split the whole thing into real and imaginary parts. So I can't show it satisfies the CRE.

    Anyone know, or should I not be using the CRE?

  2. jcsd
  3. Dec 9, 2008 #2
    Try multiplying the numerator and denominator of your expression by the complex conjugate of the denominator (that is, x-1-i y). What happens?
  4. Dec 9, 2008 #3
    Ah thanks :)

    1. The problem statement, all variables and given/known data

    Determine the largest subset of C (complex numbers) on which the following function is analytic, and compute its derivative.

    2. Relevant equations


    I'm trying to compute it's derivative right now, but how would I find its largest subset?
  5. Dec 9, 2008 #4
    First find where the function is not analytic.

    The same question was posed at https://www.physicsforums.com/showthread.php?t=278123
  6. Dec 9, 2008 #5
    so by that do u mean it is no analytic when z = 1?
  7. Dec 9, 2008 #6
    Apparantly it can be dont the same way as with real numbers, but I don't know how to do that either lol..
  8. Dec 9, 2008 #7
    bump before bed
  9. Dec 9, 2008 #8


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    Homework Helper

    The only point in question is z=1. The best outcome there would be that it's a removable singularity. It's not. Can you show that it's not?
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