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Math Help - Two Dimensional Laplace Equation

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    Two Dimensional Laplace Equation

    Given,
    u_xx + u_yy = 0
    We know by the Cauchy-Riemann equations that if f(z) is analytic on D then Re{f(x)} and Im{f(z)} are solutions to the Laplace equation on D.

    Is the converse true? And furthermore, if the converse is true can we give an explicit formula to solve Laplace equation satisfing the Dirichlet problem?
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    Quote Originally Posted by ThePerfectHacker View Post
    We know by the Cauchy-Riemann equations that if f(z) is analytic on D then Re{f(x)} and Im{f(z)} are solutions to the Laplace equation on D. Is the converse true?
    Not quite sure of what you are asking. Is this it?

    There is rather famous example: u(x,y)=x^2 – y^2 and v(x,y)=2xy.
    Both are harmonic, because they are the standard u+iv decomposition of z^2.
    Moreover, v is the harmonic conjugate of u. BUT u is not the harmonic conjugate of v, because 2xy+i(x^2 - y^2) is nowhere analytic.
    Last edited by Plato; May 13th 2007 at 02:19 PM. Reason: correction
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    Quote Originally Posted by Plato View Post
    Not quite sure of what you are asking. Is this it?
    No.

    If f(z) is analytic then Re{f(z)} and Im{f(z)} are harmonic.

    Converse (not exactly),

    If u(x,y) is a solution to the 2 Dimensional Laplace Equation (harmonic), then u = Re{f(z)} or Im{f(z)} for some analytic complex function f(z).
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    Quote Originally Posted by ThePerfectHacker View Post
    If u(x,y) is a solution to the 2 Dimensional Laplace Equation (harmonic), then u = Re{f(z)} or Im{f(z)} for some analytic complex function f(z).
    Isnít that exactly the counterexample I gave you.
    Let u(x,y)=2xy and v(x,y)=x^2 Ė y^2.
    Both are harmonic! BUT f(z)=2xy+i(x^2 - y^2) is nowhere analytic.
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    Quote Originally Posted by Plato View Post
    Isnít that exactly the counterexample I gave you.
    Let u(x,y)=2xy and v(x,y)=x^2 Ė y^2.
    Both are harmonic! BUT f(z)=2xy+i(x^2 - y^2) is nowhere analytic.
    Yes, thank you.
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