Andreotti–Norguet formula

The Andreotti–Norguet formula, first introduced by Aldo Andreotti and François Norguet (1964, 1966),^[1] is a higher–dimensional analogue of Cauchy integral formula for expressing the derivatives of a holomorphic function. Precisely, this formula express the value of the partial derivative of any multiindex order of a holomorphic function of several variables,^[2] in any interior point of a given bounded domain, as a hypersurface integral of the values of the function on the boundary of the domain itself. In this respect, it is analogous and generalizes the Bochner–Martinelli formula,^[3] reducing to it when the absolute value of the multiindex order of differentiation is 0.^[4] When considered for functions of n = 1 complex variables, it reduces to the ordinary Cauchy formula for the derivative of a holomorphic function:^[5] however, when n > 1, its integral kernel is not obtainable by simple differentiation of the Bochner–Martinelli kernel.^[6]

Historical note

The Andreotti–Norguet formula was first published in the research announcement (Andreotti & Norguet 1964, p. 780):^[7] however, its full proof was only published later in the paper (Andreotti & Norguet 1966, pp. 207–208).^[8] Another, different proof of the formula was given by Martinelli (1975).^[9] In 1977 and 1978, Lev Aizenberg gave still another proof and a generalization of the formula based on the Cauchy–Fantappiè–Leray kernel instead on the Bochner–Martinelli kernel.^[10]

The Andreotti–Norguet integral representation formula

Notation

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,
• ${\displaystyle \zeta ,z\in \mathbb {C} ^{n}}$ are complex vectors,
• ${\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})\in \mathbb {N} ^{n}}$ is a multiindex whose absolute value is |α|,
• ${\displaystyle D\subset \mathbb {C} ^{n}}$ 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: $${\displaystyle \partial ^{\alpha }f={\frac {\partial ^{|\alpha |}f}{\partial z_{1}^{\alpha _{1}}\cdots \partial z_{n}^{\alpha _{n}}}}.}$$

The Andreotti–Norguet kernel

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 ${\displaystyle I=(1,\dots ,1)\in \mathbb {N} ^{n}}$ and $${\displaystyle d{\bar {\zeta }}^{\alpha +I}[j]=d{\bar {\zeta }}_{1}^{\alpha _{1}+1}\land \cdots \land d{\bar {\zeta }}_{j-1}^{\alpha _{j+1}+1}\land d{\bar {\zeta }}_{j+1}^{\alpha _{j-1}+1}\land \cdots \land d{\bar {\zeta }}_{n}^{\alpha _{n}+1}}$$

The integral formula

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

See also

• Bergman–Weil formula