# Cauchy-Riemann equations (polar coordinates)

Suppose $A$ is an open set in $\u2102$ and $f(z)=f(r{e}^{i\theta})=u(r,\theta )+iv(r,\theta ):A\subset \u2102\to \u2102$ 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:

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

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

which are called *Cauchy-Riemann equations ^{}* in polar form.

Title | Cauchy-Riemann equations (polar coordinates^{}) |
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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 |