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Homework Help: Analytic functions

  1. Aug 17, 2014 #1
    1. The problem statement, all variables and given/known data
    Given v(x,y) find [itex]f(z) = u(x,y) +iv(x,y)[/itex]
    v(x,y) = 3y -2(x^2 - y^2) +(x) / (x^2 + y^2)

    3. The attempt at a solution

    Using Cauchy Riemann relations I've found

    [itex]dv/dx = -4x + (x^2+y^2)-1) +2x^2(x^2+y^2)-2 = -du/dx[/itex]

    Now integrate that with respect to y to find u

    But i'm not too sure how to integrate the fractions partially.

    Also I've found [itex]dv/dy = 3 +4y -2yx/(x^2 + y^2)[/itex]
  2. jcsd
  3. Aug 17, 2014 #2


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    Where is the problem? You have to find a function u, that, when you calculate the derivative with respect to x, has to have some specific form, and for the derivative with respect to y you get another known expression (check your relation(s)).
    To find u, you can use an integration with respect to x or y, respectively.
  4. Aug 17, 2014 #3
    Hi thanks for the reply. Does that mean you only have to find dv/dy and integrate with respect to x? You don't need to use dv/dx?

    Also as for the integration how is 2yx/(x^2 + y^2) integrated with respect to x?

    Thanks again.
  5. Aug 17, 2014 #4

    Ray Vickson

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    You need both ##\partial u/\partial x = \partial v/ \partial y## and ##\partial u/ \partial y = - \partial v / \partial x##.
    Last edited: Aug 17, 2014
  6. Aug 17, 2014 #5
    Thanks for the help off everyone. Last thing could anyone tell me how 2yx/(x^2 + y^2) is integrated with respect to x?
  7. Aug 17, 2014 #6


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    Staff: Mentor

    Hint: look at the derivative of the denominator. Do you see some nice substitution?
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