Andreotti–Norguet formula

The Andreotti–Norguet formula, first introduced by ,For a brief historical sketch, see the "historical section" of the present entry. 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,Partial derivatives of a holomorphic function of several complex variables are defined as partial derivatives respect to its complex arguments, i.e. as Wirtinger derivatives. 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,See , , and . reducing to it when the absolute value of the multiindex order of differentiation is .As remarked in and . When considered for functions of complex variables, it reduces to the ordinary Cauchy formula for the derivative of a holomorphic function:As remarked by . however, when , its integral kernel is not obtainable by simple differentiation of the Bochner–Martinelli kernel.See the remarks by and .

Historical note

The Andreotti–Norguet formula was first published in the research announcement :As correctly stated by and . cites only the later work which, however, contains the full proof of the formula. however, its full proof was only published later in the paper .See . Another, different proof of the formula was given by .According to , , and , who does not describe his results in this reference, but merely mentions them. 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.See , , the references cited in those sources and the brief remarks by and by : each of these works gives Aizenberg's proof.

The Andreotti–Norguet integral representation formula

Notation The notation adopted in the following description of the integral representation formula is the one used by and by : the notations used in the original works and in other references, though equivalent, are significantly different.Compare, for example, the original ones by and those used by , also briefly described in reference . Precisely, it is assumed that

is a fixed natural number, are complex vectors, is a multiindex whose absolute value is , is a bounded domain whose closure is , is the function space of functions holomorphic on the interior of and continuous on its boundary . the iterated Wirtinger derivatives of order of a given complex valued function are expressed using the following simplified notation:

The Andreotti–Norguet kernel

For every multiindex , the Andreotti–Norguet kernel is the following differential form in of bidegree :

where and

The integral formula

For every function , every point and every multiindex , the following integral representation formula holds

Andreotti–Norguet formula

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