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I Find a harmonic function with imposed restrictions

  1. Oct 1, 2016 #1
    I am trying to find a harmonic function based on the conditions imposed in the images. I see how one can make an Ansatz that ## \phi(x,y) = xy + \psi(x,y)## and can arrive at the solution given by ensuring the function satisfies the given conditions. But is there a more systematic method to solving these types of questions? In the hint, it says to use ##z^2## and it may be clear that ##z^2 = x^2 + y^2 + 2ixy##, and that the imaginary part recovers a form similar to the boundary conditions, but is there something more general one can do with this hint?

    Any suggestions on how to approach problems of this sort?
     

    Attached Files:

  2. jcsd
  3. Oct 1, 2016 #2
    As another example, I have attached another image. Once again, I can make the educated guess the solution has a term with ##sin(4\theta)## in the solution, but this is simply a guess I am making based on the conditions (i.e. ##\phi = 0## at ##\theta = 0, \frac {\pi}{4}##). The solution says ##\phi(x,y) = Im(z^4) = r^4 sin(4\theta) ##. The ##r^4 sin(4\theta) ## term makes sense to me and I see how it equals ## Im(z^4) ##, but my first guess/answer would not be something of the form ## Im(z^4) ##. Any idea on how to approach this problem with a better method?
     

    Attached Files:

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