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Math Help - Harmonic conjugate function

  1. #1
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    Harmonic conjugate function

    u= ln(x2+y2) is a harmonic function

    find a harmonic conjugate function v and the analytic function f(z) ?

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  2. #2
    Super Member
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    Try and be more precise with your notation. That's squares right? So first look at this thread:

    http://www.mathhelpforum.com/math-he...xvariable.html

    Then see if you can understand what I'm doing (lazy-style) with the following Mathematica code:

    Code:
    In[43]:=
    u[x_, y_] = Log[x^2 + y^2]; 
    v[x_, y_] = FullSimplify[
       Integrate[D[u[x, y], x], y] - 
        Integrate[D[Integrate[D[u[x, y], x], 
            y], x] + D[u[x, y], y], x]]
    FullSimplify[D[u[x, y], x] - 
       D[v[x, y], y]]
    FullSimplify[D[v[x, y], x] + 
       D[u[x, y], y]]
    
    Out[44]=
    2*ArcTan[y/x]
    
    Out[45]=
    0
    
    Out[46]=
    0
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  3. #3
    MHF Contributor

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    Quote Originally Posted by nice rose View Post
    u= ln(x2+y2) is a harmonic function

    find a harmonic conjugate function v and the analytic function f(z) ?
    Any analytic function u(x,y)+ v(x,y)i must satisfy the "Cauchy-Riemann" equations: \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y} and [tex]\frac{\partial v}{\partial x}= -\frac{\partial u}{\partial y}.

    The real and imaginary parts of any analytic function are harmonic functions and "v" is called the "harmonic conjugate" of u and vice versa.

    Since \frac{\partial ln(x^2+ y^2}{\partial x}= \frac{2x}{x^2+ y^2} and \frac{\partial ln(x^2+ y^2}{\partial y}= \frac{2y}{x^2+ y^2}, the Cauchy-Riemann equations become \frac{\partial v}{\partial y}= \frac{2x}{x^2+ y^2} and \frac{\partial v}{\partial x}= -\frac{2y}{x^2+ y^2}.

    The integral \int \frac{A}{B+ y^2}dy= \frac{A}{B}\int \frac{1}{1+ \left(\frac{y}{\sqrt{B}}\right)^2} dy so the substitution u= \frac{y}{\sqrt{B}} gives an "arctan" integral.
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