# Cauchy integral formula in several variables

Let $D=D_{1}\times\ldots\times D_{n}\subset{\mathbb{C}}^{n}$ be a polydisc.

###### Theorem.

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

 $f(z_{1},\ldots,z_{n})=\frac{1}{{(2\pi i)}^{n}}\int_{\partial D_{1}}\cdots\int_% {\partial D_{n}}\frac{f(\zeta_{1},\ldots,\zeta_{n})}{(\zeta_{1}-z_{1})\ldots(% \zeta_{n}-z_{n})}d\zeta_{1}\ldots d\zeta_{n}.$

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 $\partial D_{1}\times\ldots\times\partial D_{n}$.

## 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 CauchyIntegralFormulaInSeveralVariables 2013-03-22 15:33:46 2013-03-22 15:33:46 jirka (4157) jirka (4157) 7 jirka (4157) Theorem msc 32A07 msc 32A10