# Cauchy integral formula in several variables

Let $D={D}_{1}\times \mathrm{\dots}\times {D}_{n}\subset {\u2102}^{n}$ be a polydisc.

###### Theorem.

Let $f$ be a function continuous in $\overline{D}$ (the closure of $D$). Then $f$ is holomorphic (http://planetmath.org/HolomorphicFunctionsOfSeveralVariables) in $D$ if and only if for all $z\mathrm{=}\mathrm{(}{z}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{z}_{n}\mathrm{)}\mathrm{\in}D$ we have

$$f({z}_{1},\mathrm{\dots},{z}_{n})=\frac{1}{{(2\pi i)}^{n}}{\int}_{\partial {D}_{1}}\mathrm{\cdots}{\int}_{\partial {D}_{n}}\frac{f({\zeta}_{1},\mathrm{\dots},{\zeta}_{n})}{({\zeta}_{1}-{z}_{1})\mathrm{\dots}({\zeta}_{n}-{z}_{n})}\mathit{d}{\zeta}_{1}\mathrm{\dots}\mathit{d}{\zeta}_{n}.$$ |

As in the case of one variable this theorem can be in fact used as a definition of holomorphicity. Note that when $n>1$ then we are no longer integrating over the boundary of the polydisc but over the distinguished boundary, that is over $\partial {D}_{1}\times \mathrm{\dots}\times \partial {D}_{n}$.

## 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 | Cauchy integral formula in several variables |
---|---|

Canonical name | CauchyIntegralFormulaInSeveralVariables |

Date of creation | 2013-03-22 15:33:46 |

Last modified on | 2013-03-22 15:33:46 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 7 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 32A07 |

Classification | msc 32A10 |