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$f$ isholomorphic (http://planetmath.org/HolomorphicFunctionsOfSeveralVariables) in $D$if and only iffor 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 adefinition of holomorphicity. Note that when $n>1$ then we are no longerintegrating over the boundary of the polydisc but over thedistinguished boundary, that is over $\partial D_1\times \mathrm{\dots }\times \partial D_n$.