# generalized Cauchy integral formula

###### Theorem.

Let $U\mathrm{\subset}\mathrm{C}$ be a domain with ${C}^{\mathrm{1}}$ boundary. Let $f\mathrm{:}U\mathrm{\to}\mathrm{C}$ be a ${C}^{\mathrm{1}}$ function that is ${C}^{\mathrm{1}}$ up to the boundary. Then for $z\mathrm{\in}U\mathrm{,}$

$$f(z)=\frac{1}{2\pi i}{\int}_{\partial U}\frac{f(w)}{w-z}\mathit{d}w-\frac{1}{2\pi i}{\int}_{U}\frac{\frac{\partial f}{\partial \overline{z}}(w)}{w-z}\mathit{d}\overline{w}\wedge dw.$$ |

Note that ${C}^{1}$ up to the boundary means that the function and the derivative^{} extend to be continuous
functions^{} on the closure of $U.$ The theorem follows from Stokes’ theorem. When $f$ is holomorphic,
then the second term is zero and this is the classical Cauchy integral formula^{}.

## References

- 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | generalized Cauchy integral formula |
---|---|

Canonical name | GeneralizedCauchyIntegralFormula |

Date of creation | 2013-03-22 17:46:41 |

Last modified on | 2013-03-22 17:46:41 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 30E20 |

Synonym | generalized Cauchy formula |