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Unable to show that functions are harmonic.

  1. Feb 11, 2009 #1
    1. The problem statement, all variables and given/known data

    If the functions [tex]u(x,y)[/tex] and [tex]v(x,y)[/tex] have continuous second partial derivatives and they satisfy the Cauchy-Riemann equations. Show that [tex]u(x,y)[/tex] and [tex]v(x,y)[/tex] are harmonic functions.

    2. Relevant equations

    The Cauchy-Riemann equations are given: [tex]\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}[/tex] and [tex]\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}[/tex]

    And functions are harmonic if [tex]\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2} = 0[/tex] (Laplace equation)

    3. The attempt at a solution

    I've been stuck with this one for couple of hours now, and I really can't get much out of it. The only thing that's gone through my head is to differentiate the Cauchy-Riemann equations once more and trying to arrange the terms so that the Laplace equation is satisfied. But to no avail.
     
  2. jcsd
  3. Feb 11, 2009 #2

    Dick

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    Are you forgetting that u,x,y=u,y,x (the commas indicate partial derivatives). I.e. the derivative is equal regardless of the order of differentiation.
     
  4. Feb 11, 2009 #3
    No, I'm not. But you're tip doesn't ring a bell.

    Edit: I'll take that back! Got it now! Thanks for the help! How can I be so stupid...
     
  5. Feb 11, 2009 #4

    Dick

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    Differentiate the first equation with respect to x and the second one with respect to y. The first equation has u,xx in it and the second one has -u,yy. And they are equal to the same thing. So they are equal to each other.
     
  6. Feb 11, 2009 #5
    I solved it just before you replied. I can't believe I couldn't see that. Thanks a bunch!
     
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