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Determine whether functions are harmonic

  1. Apr 6, 2015 #1
    1. The problem statement, all variables and given/known data

    Determine whether or not the following functions are harmonic:

    [itex]u = z + \bar{z} [/itex]

    [itex]u = 2z\bar{z} [/itex]

    2. Relevant equations

    [itex]z = u(x,y) + v(x,y)i [/itex]

    [itex]\bar{z} = u(x,y) - v(x,y)i [/itex]

    A function is harmonic if Δu = 0.

    3. The attempt at a solution

    [itex]Δu = Δz +Δ \bar{z} = u_{xx} + v_{xx} + u_{yy} + v_{yy} + u_{xx} - v_{xx} + u_{yy} + -v_{yy} = 2u_{xx} + 2u_{yy}

    [/itex]

    [itex]u = 2z\bar{z} = 2[u(x,y) + v(x,y)i][u(x,y) - v(x,y)i] = 2[u^2(x,y) - v^2(x,y)][/itex]

    [itex]Δu = 2[2uu_{xx} + 2uu_{yy} - 2vv_{xx} - 2vv_{yy}]

    [/itex]
     
    Last edited: Apr 6, 2015
  2. jcsd
  3. Apr 6, 2015 #2

    Dick

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    I don't see why you are struggling with this. ##z=x+iy##. If ##u=z+\bar{z}## then ##u(x,y)=2x##. Is that harmonic?
     
  4. Apr 6, 2015 #3
    I wanted to use the more general case. To be honest, I just wanted to check my work.

    If it's not zero, then it's not harmonic.
     
  5. Apr 6, 2015 #4

    Dick

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    I'm not sure what you are saying here. Don't do the general case. Just do these two special cases. What about those?
     
  6. Apr 6, 2015 #5
    Sorry. It was my mistake. For some reason I wanted to generalize to a function f(z).

    Here, the first is harmonic. Δ(2x) = 0 and Δ(2x2+2y2) = 4 + 4 = 8.
     
  7. Apr 6, 2015 #6

    Dick

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    ##(x+iy)(x-iy)## is not equal to ##x^2-y^2##.
     
  8. Apr 6, 2015 #7
    Corrected.
     
  9. Apr 6, 2015 #8

    Dick

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    That's better.
     
  10. Apr 6, 2015 #9
    Thanks again.
     
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