Cauchy-Riemann equations (polar coordinates)

Suppose $A$ is an open set in $ℂ$ and $f(z)=f(r{e}^{i\theta })=u(r,\theta )+iv(r,\theta ):A\subset ℂ\to ℂ$ is a function. If the derivative of $f(z)$ exists at $z_0=(r_0,{\theta }_0)$. Then the functions $u$, $v$ at $z_0$ satisfy:

${\displaystyle \frac{\partial u}{\partial r}}$ $=$ ${\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial v}{\partial \theta }}$

${\displaystyle \frac{\partial v}{\partial r}}$ $=$ $-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u}{\partial \theta }}$

which are called Cauchy-Riemann equations in polar form.

Title	Cauchy-Riemann equations (polar coordinates)
Canonical name	CauchyRiemannEquationspolarCoordinates
Date of creation	2013-03-22 14:03:58
Last modified on	2013-03-22 14:03:58
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	8
Author	Daume (40)
Entry type	Definition
Classification	msc 30E99
Related topic	TangentialCauchyRiemannComplexOfCinftySmoothForms
Related topic	ACRcomplex