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Math Help - Laplace equation

  1. #1
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    Laplace equation

    How can I find infinitely many functions u:R^2 \rightarrow R that solve the Laplace equation? I think I need to find analytic functions f in open subset U \subset C, but I don't know how to use the Cauchy Riemann equations here to help me. Someone can help please.
    \frac{\delta^2 u}{\delta x^2}+\frac{\delta^2 u}{\delta y^2}=0
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  2. #2
    Super Member Rebesques's Avatar
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    You can find loads, say x^2-y^2, \ {\rm e}^x{\rm cos}y, \ \rm log(x^2+y^2)^{1/2}
    and try superposition.
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    What about  \{ax^2-ay^2 | a \in \mathbb{R} \} ?
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  4. #4
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    Quote Originally Posted by jackie View Post
    How can I find infinitely many functions u:R^2 \rightarrow R that solve the Laplace equation? I think I need to find analytic functions f in open subset U \subset C, but I don't know how to use the Cauchy Riemann equations here to help me. Someone can help please.
    \frac{\delta^2 u}{\delta x^2}+\frac{\delta^2 u}{\delta y^2}=0
    Remember that every analytic function (in the open set \Omega ) can be seen as f= u +iv where u,v: \Omega \longrightarrow \mathbb{R} are such that u,v \in C^1(\Omega ) and satisfy \frac{ \partial u}{ \partial x} = \frac{ \partial v}{ \partial y} and \frac{ \partial u}{ \partial y} = - \frac{ \partial v}{ \partial x} (with u= u(x,y) and v=v(x,y)). Now assume that u,v \in C^2 (\Omega) (if you haven't proved that the real and imaginary part of an anlytic function are of C^{\infty} class) and using the C-R equations we get:

    \frac{ \partial^2 u}{ \partial x^2} = \frac{ \partial^2 v}{ \partial x \partial y}

    \frac{ \partial^2 u}{ \partial y^2} = - \frac{ \partial^2 v}{ \partial y \partial x}

    And since u,v \in C^2 (\Omega ) we have that the mixed partial derivatives of v are equal and so, adding the last two equations together we get that u satisfies the Laplace equation in \Omega (the same argument tells you that v also satisfies Laplace in \Omega ). And so the real and imaginary part of an analytic function are solutions to the Laplace equation (ie. they're harmonic functions)
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