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Math Help - Complex Conjugate

  1. #1
    Member Jason Bourne's Avatar
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    Complex Conjugate

    I have the following question:

    Given that

    u(x,y) = x~cosx~coshy + y~sinx~sinhy

    Find a funvtion v(x,y) such that

    w(z) = u(x,y) + iv(x,y)

    is an analytic function of z. Find w(z) explicitly in terms of z.
    -----------------------------------------------

    I know how to do these type of questions by using Cauchy-Riemann equations but I was wondering if there was a way of making this one simpler as it becomes very tedious. Is there an identity I can use to make the process much simpler?
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  2. #2
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    Quote Originally Posted by Jason Bourne View Post
    u(x,y) = x~cosx~coshy + y~sinx~sinhy
    You need to find u(x,y)+iv(x,y) so that \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}.

    Can you solve that?
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  3. #3
    Member Jason Bourne's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    You need to find u(x,y)+iv(x,y) so that \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}.

    Can you solve that?
    Yes I know how to solve it, but the problem involved lots of calculus and it got all messed up and way too long, so I was thinking that there must be an identity so that I could simplify the problem and make it more bearable so that im not insane by the end of the question.
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