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Thread: Complex function of complexvariable

  1. #1
    Member Ruun's Avatar
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    Complex function of complexvariable

    If I am given the function $\displaystyle u(x,y)$ wich could be the real part of an analytic complex function of complex variable, and $\displaystyle z=x+iy$, how can I compute a function $\displaystyle f(z)=u(x,y)+iv(x,y)$? I have to apply the Cauchy Riemann conditions ( $\displaystyle u_{x}=v_{y}$ and $\displaystyle u_{y}=-v_{x}$) and integrate both equations?

    Thanks in advance
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  2. #2
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    Assume $\displaystyle u(x,y)$ is harmonic, then it has a harmonic conjugate $\displaystyle v(x,y)$ such that $\displaystyle f(x,y)=u(x,y)+i v(x,y)$ is analytic and satisfies the partials:

    $\displaystyle \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\quad \quad \frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}$

    so yes, you can partially integrate them like:

    $\displaystyle \int \partial v=\int \frac{\partial u}{\partial x}\partial y$

    and after the integration with respect to y with x held constant, we obtain something like:

    $\displaystyle v(x,y)=\int \frac{\partial u}{\partial x}\partial y+\phi(x)$

    where the constant of integration is now a function of x right? Now we use the second CR equations to solve for $\displaystyle \phi(x)$ by writing:

    $\displaystyle \frac{\partial u}{\partial y}=-\frac{\partial}{\partial x}\left\{\int \frac{\partial u}{\partial x}\partial y+\phi(x)\right\}$

    Now, solve for $\displaystyle \phi'(x)$ and integrate.
    Do it manually a few times first to get the hang of it then explain how that's the same as:

    $\displaystyle v(x,y)=\int\frac{\partial u}{\partial x}\partial y-\int\left(\frac{\partial}{\partial x}\int \frac{\partial u}{\partial x}\partial y+\frac{\partial u}{\partial y}\right)dx$

    where all the integrations are without the arbitrary function, and it looks intimidating at first sight but becomes a piece of cake after you solve a few of them. For example $\displaystyle u(x,y)=y^3-3x^2y$, then:

    $\displaystyle \int \frac{\partial u}{\partial x}\partial y=\int (-6xy)\partial y=-3xy^2$

    Now take the derivative of that with respect to x:

    $\displaystyle \frac{\partial}{\partial x}\left(\int \frac{\partial u}{\partial x}\partial y\right)=\frac{\partial}{\partial x}(-3xy^2)=-3y^2$

    and continue working the partials and integrals until you get the expression for v(x,y). Then the most general harmonic conjugate is v(x,y)+k.
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  3. #3
    Member Ruun's Avatar
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    Yes, by "integrating the equations" I mean what you have done, taking in count that function $\displaystyle \phi (x)$ with the partial respect to $\displaystyle y$, not just integrating both.

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