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I Complex Analysis Harmonic functions

  1. Nov 7, 2016 #1
    Suppose u(x,y) and v(x,y) are harmonic on G and c is an element of R. Prove u(x,y) + cv(x,y) is also harmonic.

    I tried using the Laplace Equation of Uxx+Uyy=0

    I have:
    du/dx=Ux
    d^2u/dx^2=Uxx

    du/dy=Uy
    d^2u/dy^2=Uyy

    dv/dx=cVx
    d^2v/dx^2=cVxx

    dv/dy=cVy
    d^2v/dy^2=cVyy
    I'm not really sure how to prove these are harmonic...am I missing a relationship?
     
  2. jcsd
  3. Nov 7, 2016 #2

    FactChecker

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    Define w(x,y) = u(x,y) + cv(x,y) and calculate wxx + wyy. Basic properties of the partial derivative should give you the answer.
     
  4. Nov 8, 2016 #3
    Wow, that's a really good idea. I was trying to do the harmonic conjugate but was getting nowhere. Thank you!
     
  5. Nov 10, 2016 #4

    mathwonk

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    do you understand what it means to say that differentiation is a linear operator?
     
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