# Cauchy-Riemann equations (polar coordinates)

Suppose $A$ is an open set in $\mathbb{C}$ and $f(z)=f(re^{i\theta})=u(r,\theta)+iv(r,\theta):A\subset\mathbb{C}\to\mathbb{C}$ 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:

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

which are called Cauchy-Riemann equations in polar form.

Title Cauchy-Riemann equations (polar coordinates) CauchyRiemannEquationspolarCoordinates 2013-03-22 14:03:58 2013-03-22 14:03:58 Daume (40) Daume (40) 8 Daume (40) Definition msc 30E99 TangentialCauchyRiemannComplexOfCinftySmoothForms ACRcomplex