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Harmonic functions

  1. Nov 26, 2014 #1
    If $$f(z)=u(x,y)+iv(x,y)$$ is analytic in a domain D, then both u and v satisfy Laplace's equations
    $$\nabla^2 u=u_{xx} + u_{yy}=0$$
    $$\nabla^2 v=v_{xx} + v_{yy}=0$$

    and u and v are called harmonic functions.

    My question is whether or not this goes both ways. If you have two functions u and v which satisfy the Laplace equations are they the real and imaginary parts of an analytic function?
     
  2. jcsd
  3. Nov 26, 2014 #2

    ShayanJ

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    Just take the real part of one function and the imaginary part of another function. They satisfy Laplace's equation but aren't the real and imaginary parts of a function because they don't satisfy the Cauchy-Riemann conditions.
     
  4. Nov 28, 2014 #3

    mathwonk

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    A Shyan said, the answer to your question is no, since the real part of a holomorphic function determines the imaginary part. But every harmonic function is locally the real part of a holomorphic function, although not necessarily globally, due to branching behavior that may occur only in the imaginary part. E.g. log(|z|), defined everywhere but z=0, is the real part of log(z), but the imaginary part of log(z), a multiple of arg(z), is only defined locally near non zero values of z.
     
  5. Nov 28, 2014 #4

    WWGD

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    In yet another way, the Real part of a holomorphic function can only have ( up to a difference by a constant) one
    Complex counterpart, so the odds are that two harmonic functions are respectively the Real and Imaginary part of a holomorphic function.
     
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