Cauchy-Riemann equations (polar coordinates)CauchyRiemannEquationsPolarCoordinates2003-11-15 12:05:142005-11-06 19:24:41DefinitionCauchy-Riemann equations% this is the default PlanetMath preamble. as your knowledge
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% define commands hereSuppose $A$ 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:
\begin{eqnarray*}
\frac{\partial u}{\partial r} & = & \frac{1}{r}\frac{\partial v}{\partial \theta}\\
\frac{\partial v}{\partial r} & = & -\frac{1}{r}\frac{\partial u}{\partial \theta}
\end{eqnarray*}
which are called \emph{Cauchy-Riemann equations} in polar form.\\\\