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Laplace u => Harmonic function

  1. Sep 5, 2008 #1
    In R2, I am to find all homog. polys (deg 2) that are harmonic.

    the earlier homework included something like u = xy, show it's harmonic. EASY as pi. But I'm not really sure how to set this problem up. I understand the concept that a harmonic function will look like [tex] \nabla^{2} u = 0 [/tex], but I'm not sure how to find all polys of degree 2.

    I started out doing something like

    [tex] u = a_{2}x^{2}_{1} + a_{1}x_{1} + a_{0} + b_{2}x^{2}_{2} + b_{1}x_{2} + b_{0}+... [/tex]

    and taking the partial with respect to each xi but that's not getting me very far.

    Any suggestions?
  2. jcsd
  3. Sep 5, 2008 #2

    Ben Niehoff

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    Gold Member

    If you are in R^2, then you only need x and y. The general 2nd-degree polynomial is simply

    [tex]Ax^2 + Bxy + Cy^2 + Dx + Ey + F[/tex]
  4. Sep 5, 2008 #3
    Thank you. I ended up with f(x,y) = Ax2 + Bx + C - Ay2 + Ey + F covers harmonic polys in R2.
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