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'Cauchy-Riemann equations (polar coordinates)'
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| Title of object: |
Cauchy-Riemann equations (polar coordinates) |
| Canonical Name: |
CauchyRiemannEquationsPolarCoordinates |
| Type: |
Definition |
| Created on: |
2003-11-15 12:05:14 |
| Modified on: |
2003-11-15 12:05:14 |
| Classification: |
msc:30E99 |
Revision comment (for changes between this and next version):
| Changes for correction #2825 ('undefined terms'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
Suppose $A$ is an open set in $\mathbb{C}$ and $f=u(r,\theta)+iv(r,\theta): A\subset\mathbb{C} \to \mathbb{C}$ 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. |
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