The Andreotti–Norguet formula, first introduced by , is a higher–dimensional analogue of
Cauchy integral formula for expressing the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
. Precisely, this formula express the value of the
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
of any
multiindex
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
of a
holomorphic function of several variables, in any
interior point of a given
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
, 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 In mathematics, the Bochner–Martinelli formula is a generalization of the Cauchy integral formula to functions of several complex variables, introduced by and .
History
Bochner–Martinelli kernel
For , in \C^n the Bochner–Martinelli ke ...
, reducing to it when the absolute value of the multiindex order of differentiation is . When considered for functions of complex variables, it reduces to the ordinary Cauchy formula for the derivative of a holomorphic function: however, when , 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 : however, its full proof was only published later in the paper . Another, different proof of the formula was given by . In 1977 and 1978,
Lev Aizenberg
Lev may refer to:
Common uses
*Bulgarian lev, the currency of Bulgaria
*an abbreviation for Leviticus, the third book of the Hebrew Bible and the Torah
People and fictional characters
*Lev (given name)
*Lev (surname)
Places
*Lev, Azerbaijan, a ...
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 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
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
,
*
are
complex vectors,
*
is a
multiindex
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
whose
absolute value is ,
*
is a bounded domain whose
closure is ,
* is the
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
of functions holomorphic on the
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
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 mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
in of bidegree :
where
and
The integral formula
For every function , every point and every multiindex , the following integral representation formula holds
See also
*
Bergman–Weil formula
Notes
References
*, revised translation of the 1990 Russian original.
*.
*.
*.
* , .
*.
*.
*, (ebook).
*. Collection of articles dedicated to Giovanni Sansone on the occasion of his eighty-fifth birthday.
*. The notes form a course, published by the
Accademia Nazionale dei Lincei
The Accademia dei Lincei (; literally the "Academy of the Lynx-Eyed", but anglicised as the Lincean Academy) is one of the oldest and most prestigious European scientific institutions, located at the Palazzo Corsini on the Via della Lungara in Rom ...
, held by Martinelli during his stay at the Accademia as "''Professore Linceo''".
{{DEFAULTSORT:Andreotti-Norguet formula
Theorems in complex analysis
Several complex variables