# real and imaginary parts of contour integral

If $f(z)$ is continuous^{} on the contour $\gamma $ of the complex plane^{} and

$$z=x+iy\mathit{\hspace{1em}}(x,y\in \mathbb{R}),\mathrm{\Re}f(z)=u(x,y),\mathrm{\Im}f(z)=v(x,y),$$ |

then the contour integral of $f(z)$ along $\gamma $ may be expressed via two real path integrals as

$${\int}_{\gamma}f(z)\mathit{d}z={\int}_{\gamma}(udx-vdy)+i{\int}_{\gamma}(vdx+udy).$$ |

Title | real and imaginary parts of contour integral |
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Canonical name | RealAndImaginaryPartsOfContourIntegral |

Date of creation | 2013-03-22 19:14:32 |

Last modified on | 2013-03-22 19:14:32 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Result |

Classification | msc 26B20 |

Classification | msc 30E20 |

Classification | msc 30A99 |

Related topic | ContourIntegral |

Related topic | PathIntegral |

Related topic | RealPartSeriesAndImaginaryPartSeries |