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[parent] Cauchy-Riemann equations (polar coordinates) (Definition)

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.



"Cauchy-Riemann equations (polar coordinates)" is owned by Daume.
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Cross-references: polar form, Cauchy-Riemann equations, derivative, function, open set

This is version 5 of Cauchy-Riemann equations (polar coordinates), born on 2003-11-15, modified 2005-11-06.
Object id is 5423, canonical name is CauchyRiemannEquationsPolarCoordinates.
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AMS MSC: 30E99 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Miscellaneous)
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