Cauchy integral formula in several variables

Let $D=D_1\times \mathrm{\dots }\times D_n\subset {ℂ}^n$ 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

$$f(z_1,\mathrm{\dots },z_n)=\frac{1}{{(2\pi i)}^n}{\int }_{\partial D_1}\mathrm{\cdots }{\int }_{\partial D_n}\frac{f({\zeta }_1,\mathrm{\dots },{\zeta }_n)}{({\zeta }_1-z_1)\mathrm{\dots }({\zeta }_n-z_n)}𝑑{\zeta }_1\mathrm{\dots }𝑑{\zeta }_n.$$

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 $\partial D_1\times \mathrm{\dots }\times \partial D_n$.

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