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,
$\zeta ,z\in \mathbb {C} ^{n}$ are complex vectors,
$\alpha =(\alpha _{1},\dots ,\alpha _{n})\in \mathbb {N} ^{n}$ is a multiindex whose absolute value is $| α |$ ,
$D\subset \mathbb {C} ^{n}$ 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 $\partialD$ .
the iterated Wirtinger derivatives of order $α$ of a given complex valued function $f \in A (D)$ are expressed using the following simplified notation: $\partial ^{\alpha }f={\frac {\partial ^{|\alpha |}f}{\partial z_{1}^{\alpha _{1}}\cdots \partial z_{n}^{\alpha _{n}}}}.$

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

The integral formula

Theorem 1 (Andreotti and Norguet). For every function $f \in A (D)$ , every point $z \in D$ and every multiindex $α$ , the following integral representation formula holds $\partial ^{\alpha }f(z)=\int _{\partial D}f(\zeta )\omega _{\alpha }(\zeta ,z).$

Andreotti–Norguet formula

Historical note

The Andreotti–Norguet integral representation formula

Notation

The Andreotti–Norguet kernel

The integral formula

See also

Notes

References

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