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Thread: Cauchy-Riemann equations in polar form

  1. #1
    MHF Contributor arbolis's Avatar
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    Cauchy-Riemann equations in polar form

    I'm totally stuck on this exercise. I'd like a hint rather than a full answer.

    Let $\displaystyle r$ and $\displaystyle \theta$ be the polar coordinates in the plane (complex I guess?), let $\displaystyle f(z)$ be a function of the complex plane in itself. (I guess they mean $\displaystyle f:\mathbb{C} \to \mathbb{C}$.)
    Using the fact that $\displaystyle f(re^{i\theta})=R(r,\theta)e^{i\Theta (r, \theta)}$ in which $\displaystyle R(r,\theta)$ and $\displaystyle \Theta (r,\theta)$ are real differentiable functions of $\displaystyle r$ and $\displaystyle \theta$, show that the Cauchy-Riemann equations in polar coordinates are written as $\displaystyle \frac{\partial R}{\partial r}=\frac{R \partial \Theta}{r \partial \theta}$ and $\displaystyle \frac{\partial R}{r \partial \theta}=-R \frac{\partial \Theta}{\partial r}$.

    I must also show another equation, but I'll ask help if I get stuck.


    I don't know how to start the problem. I know the Cauchy Riemann equations, but I don't think I should start by writing them down.
    Last edited by arbolis; Aug 20th 2009 at 02:55 PM.
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  2. #2
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    Jester's Avatar
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    Quote Originally Posted by arbolis View Post
    I'm totally stuck on this exercise. I'd like a hint rather than a full answer.

    Let $\displaystyle r$ and $\displaystyle \theta$ be the polar coordinates in the plane (complex I guess?), let $\displaystyle f(z)$ be a function of the complex plane in itself. (I guess they mean $\displaystyle f:\mathbb{C} \to \mathbb{C}$.)
    Using the fact that $\displaystyle f(re^{i\theta})=R(r,\theta)e^{i\Theta (r, \theta)}$ in which $\displaystyle R(r,\theta)$ and $\displaystyle \Theta (r,\theta)$ are real differentiable functions of $\displaystyle r$ and $\displaystyle \theta$, show that the Cauchy-Riemann equations in polar coordinates are written as $\displaystyle \frac{\partial R}{\partial r}=\frac{\partial R}{r \partial \theta}=-R \frac{\partial \Theta}{\partial r}$.

    I must also show another equation, but I'll ask help if I get stuck.


    I don't know how to start the problem. I know the Cauchy Riemann equations, but I don't think I should start by writing them down.
    I would first find differential operators linking

    $\displaystyle \frac{ \partial }{\partial x}$ and $\displaystyle \frac{ \partial }{\partial y} $ to $\displaystyle \frac{ \partial }{\partial r} $ and $\displaystyle \frac{ \partial }{\partial \theta}$

    then identify $\displaystyle u$ and $\displaystyle v$ from

    $\displaystyle u + i v = f(re^{i\theta})=R(r,\theta)e^{i\Theta (r, \theta)}$

    and then substitute everything into the CR equations and your new equations in polar form should merge.
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  3. #3
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Danny View Post
    I would first find differential operators linking

    $\displaystyle \frac{ \partial }{\partial x}$ and $\displaystyle \frac{ \partial }{\partial y} $ to $\displaystyle \frac{ \partial }{\partial r} $ and $\displaystyle \frac{ \partial }{\partial \theta}$

    then identify $\displaystyle u$ and $\displaystyle v$ from

    $\displaystyle u + i v = f(re^{i\theta})=R(r,\theta)e^{i\Theta (r, \theta)}$

    and then substitute everything into the CR equations and your new equations in polar form should merge.
    Ok thanks. By the way I made a typo in the equations, I'll edit it right now.
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Danny View Post
    I would first find differential operators linking

    $\displaystyle \frac{ \partial }{\partial x}$ and $\displaystyle \frac{ \partial }{\partial y} $ to $\displaystyle \frac{ \partial }{\partial r} $ and $\displaystyle \frac{ \partial }{\partial \theta}$

    then identify $\displaystyle u$ and $\displaystyle v$ from

    $\displaystyle u + i v = f(re^{i\theta})=R(r,\theta)e^{i\Theta (r, \theta)}$

    and then substitute everything into the CR equations and your new equations in polar form should merge.
    I did something like this for homework in my complex analysis class. The question was asked differently (but the final result was to give the polar form of the C-R equations).

    @ arbolis: Look at the attachment at your own discretion. I have the whole thing worked out...but it may be different from what you're supposed to do.
    Attached Files Attached Files
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