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Math Help - Complex Analysis, harmonic functions

  1. #1
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    Complex Analysis, harmonic functions

    Here is the problem:

    If f= u+ iv is analytic in a region, show that uv is harmonic in the region but that u^2 need not be harmonic.

    Thanks
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  2. #2
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    Quote Originally Posted by taypez View Post
    Here is the problem:

    If f= u+ iv is analytic in a region, show that uv is harmonic in the region but that u^2 need not be harmonic.

    Thanks
    f(x+iy)=u+iv
    The function is analyic, hence homolorphic.
    It satisfies the Cauchy-Riemann Equations.
    u_x=v_y
    u_y=-v_x
    Thus,
    u_xv_x+u_yv_y=0

    We need to show,
    uv
    Solves the PDE,
    f_{xx}+f_{yy}=0.

    Now,
    (uv)_{xx}=(u_xv+uv_x)_x=u_{xx}v+2u_xv_x+uv_{xx}
    (uv)_{yy}=u_{yy}v+2u_yv_y+uv_{yy}
    Add them,
    (uv)_{xx}+(uv)_{yy}=u_{xx}v+2(u_xv_x+u_yv_y)+uv_{y  y}
    But from Cauchy-Riemann Equations,
    (uv)_{xx}+(uv)_{yy}=u_{xx}v+uv_{yy}
    Maybe there is some theorem that tell the right hand side is zero. I do not know of one.
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  3. #3
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    The function is analyic, hence both u and v are harmonic.
    It satisfies the Cauchy-Riemann Equations.
    u_x=v_y & u_y=-v_x
    Now,
    (uv)_{xx}=(u_xv+uv_x)_x=u_{xx}v+2u_xv_x+uv_{xx}
    (uv)_{yy}=u_{yy}v+2u_yv_y+uv_{yy}
    Add them,
    (uv)_{xx}+(uv)_{yy}=2(u_xv_x+u_yv_y) because
    u_{xx}+ u_{yy}=0 & v_{xx}+ v_{yy}=0.

    But u_x v_x  + u_y v_y  = u_x \left( { - u_y } \right) + u_y \left( { u_x } \right) = 0
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  4. #4
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    Quote Originally Posted by Plato View Post
    The function is analyic, hence both u and v are harmonic.
    Slow down, I did not get that. You are saying that if a complex function is analytic (meaning it has a power series expansion, right?) then it must be harmonic, (satisfy the Laplace equation). So that was the extra equation you used to complete the proof?
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  5. #5
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    Quote Originally Posted by ThePerfectHacker View Post
    a complex function is analytic (meaning it has a power series expansion, right?) then it must be harmonic, (satisfy the Laplace equation).
    For the complex analyst the definition is: a function f said to be analytic at z_0 if its derivative exists at each point in some neighborhood of z_0.

    Moreover, a function u is said to be harmonic if u_{xx}  + u_{yy}  = 0, that is the Laplace equation as you have noted.

    Theorem: If f(z) = u(x,y) + iv(x,y) is analytic is the domain D, then both u(x,y) and v(x,y) are harmonic in D.
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