Andreotti–Norguet formula

The Andreotti–Norguet formula, first introduced by Template:Harvs, 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, 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, reducing to it when the absolute value of the multiindex order of differentiation is Template:Math. When considered for functions of Template:Math complex variables, it reduces to the ordinary Cauchy formula for the derivative of a holomorphic function: however, when Template:Math, its integral kernel is not obtainable by simple differentiation of the Bochner–Martinelli kernel.

Historical note

The Andreotti–Norguet formula was first published in the research announcement Template:Harv: however, its full proof was only published later in the paper Template:Harv. Another, different proof of the formula was given by Template:Harvtxt. 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.

The Andreotti–Norguet integral representation formula

Notation

The notation adopted in the following description of the integral representation formula is the one used by Template:Harvtxt and by Template:Harvtxt: the notations used in the original works and in other references, though equivalent, are significantly different. Precisely, it is assumed that

• Template:Math is a fixed natural number,
• $\zeta ,z\in {ℂ}^n$ are complex vectors,
• $\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}^n$ is a multiindex whose absolute value is Template:Math,
• $D\subset {ℂ}^n$ is a bounded domain whose closure is Template:Math,
• Template:Math is the function space of functions holomorphic on the interior of Template:Math and continuous on its boundary Template:Math.
• the iterated Wirtinger derivatives of order Template:Math of a given complex valued function Template:Math 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

Template:EquationRef For every multiindex Template:Math, the Andreotti–Norguet kernel Template:Math is the following differential form in Template:Math of bidegree Template:Math: $${\omega }_{\alpha }(\zeta ,z)=\frac{(n-1)!{\alpha }_1!\cdots {\alpha }_n!}{(2\pi i{)}^n}\sum _{j=1}^n\frac{(-1{)}^{j-1}({\overline{\zeta }}_j-{\bar{z}}_j{)}^{{\alpha }_j+1}d{\overline{\zeta }}^{\alpha +I}[j]\wedge d\zeta }{{(|z_1-{\zeta }_1{|}^{2({\alpha }_1+1)}+\cdots +|z_n-{\zeta }_n{|}^{2({\alpha }_n+1)})}^n},$$ where $I=(1,\dots ,1)\in {\mathbb{N}}^n$ and $$d{\overline{\zeta }}^{\alpha +I}[j]=d{\overline{\zeta }}_1^{{\alpha }_1+1}\wedge \cdots \wedge d{\overline{\zeta }}_{j-1}^{{\alpha }_{j+1}+1}\wedge d{\overline{\zeta }}_{j+1}^{{\alpha }_{j-1}+1}\wedge \cdots \wedge d{\overline{\zeta }}_n^{{\alpha }_n+1}$$

The integral formula

Template:EquationRef For every function Template:Math, every point Template:Math and every multiindex Template:Math, the following integral representation formula holds $${\partial }^{\alpha }f(z)=\underset{\partial D}{\int }f(\zeta ){\omega }_{\alpha }(\zeta ,z).$$

See also

• Bergman–Weil formula

References

• Template:Citation, revised translation of the 1990 Russian original.
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• Template:Citation Template:ISBN, Template:ISBN.
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• Template:Citation, Template:ISBN (ebook).
• Template:Citation. Collection of articles dedicated to Giovanni Sansone on the occasion of his eighty-fifth birthday.
• Template:Citation. The notes form a course, published by the Accademia Nazionale dei Lincei, held by Martinelli during his stay at the Accademia as "Professore Linceo".
• Template:Citation, :