Thread: Cauchy-Riemann equations in polar form

1. 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.

2. Originally Posted by arbolis
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.

3. Originally Posted by Danny
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.

4. Originally Posted by Danny
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.