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Show that f is entire when u is harmonic and v=

  1. Mar 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Assume that u is harmonic everywhere in R^2, and let
    [tex]v(x,y)=\int_0^y u_x'(x,t)dt - \int_0^x u_y'(s,0)ds[/tex]
    show that f=u+iv is entire analytic.


    2. Relevant equations
    Maybe Cauchy Riemann: [tex]u_x'=v_y'[/tex] and [tex]u_y'=-v_x'[/tex]


    3. The attempt at a solution
    I have only tried to see what happens if I use the Cauchy Riemann equations, but I get stuck right away. I am not sure how to use the fact that u is harmonic either.

    Any hints would be very appreciated.
     
  2. jcsd
  3. Mar 26, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    ux' is very strange notation. I think you mean just the derivative of u with respect to x- but that is just ux. There is no need for the ' here.

    Good. Show what you did and where you got stuck.

    What does "harmonic" mean?
     
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