Andreotti–Norguet formula

The notation adopted in the following description of the integral representation formula is the one used by Kytmanov (1995, p. 9) and by Kytmanov & Myslivets (2010, p. 20): the notations used in the original works and in other references, though equivalent, are significantly different.[11] Precisely, it is assumed that

• n > 1 is a fixed natural number,
• ζ,  z ∈ ℂⁿ are complex vectors,
• α = (α₁,...,α_n) ∈ ℕⁿ is a multiindex whose absolute value is |α|,
• D ⊂ ℂⁿ is a bounded domain whose closure is D,
• A(D) is the function space of functions holomorphic on the interior of D and continuous on its boundary ∂D.
• the iterated Wirtinger derivatives of order α of a given complex valued function f ∈ A(D) are expressed using the following simplified notation:

${\displaystyle \partial ^{\alpha }f={\frac {\partial ^{|\alpha |}f}{\partial z_{1}^{\alpha _{1}}\cdots \partial z_{n}^{\alpha _{n}}}}.}$

Definition 1. For every multiindex α, the Andreotti–Norguet kernel ω_α (ζ, z) is the following differential form in ζ of bidegree (n, n − 1):

${\displaystyle \omega _{\alpha }(\zeta ,z)={\frac {(n-1)!\alpha _{1}!\cdots \alpha _{n}!}{(2\pi i)^{n}}}\sum _{j=1}^{n}{\frac {(-1)^{j-1}({\bar {\zeta }}_{j}-{\overline {z}}_{j})^{\alpha _{j}+1}\,d{\bar {\zeta }}^{\alpha +I}[j]\land d\zeta }{\left(|z_{1}-\zeta _{1}|^{2(\alpha _{1}+1)}+\cdots +|z_{n}-\zeta _{n}|^{2(\alpha _{n}+1)}\right)^{n}}},}$

where I = (1,...,1) ∈ ℕⁿ and

${\displaystyle d{\bar {\zeta }}^{\alpha +I}[j]=d{\bar {\zeta }}_{1}^{\alpha _{1}+1}\land \cdots \land d{\bar {\zeta }}_{j-1}^{\alpha _{j+1}+1}\land d{\bar {\zeta }}_{j+1}^{\alpha _{j-1}+1}\land \cdots \land d{\bar {\zeta }}_{n}^{\alpha _{n}+1}}$

Theorem 1 (Andreotti and Norguet). For every function f ∈ A(D), every point z ∈ D and every multiindex α, the following integral representation formula holds

${\displaystyle \partial ^{\alpha }f(z)=\int _{\partial D}f(\zeta )\omega _{\alpha }(\zeta ,z).}$