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The '''Andreotti–Norguet formula''', first introduced by {{harvs|txt|first=Aldo|last=Andreotti|author1-link=Aldo Andreotti|first2=François|last2=Norguet|author2-link=François Norguet|year=1964|year2=1966}},<ref>For a brief historical sketch, see the "[[Andreotti–Norguet formula#Historical note|historical section]]" of the present entry.</ref> is a higher–dimensional analogue of [[Cauchy integral formula]] for expressing the [[derivative]]s of a [[holomorphic function]]. Precisely, this formula express the value of the [[partial derivative]] of any [[multiindex]] [[Order of derivation|order]] of a [[Several complex variables|holomorphic function of several variables]],<ref>Partial derivatives of a holomorphic function of several complex variables are defined as partial derivatives respect to its [[complex number|complex]] [[Argument of a function|arguments]], i.e. as [[Wirtinger derivatives]].</ref> in any [[interior point]] of a given [[bounded set|bounded]] [[Domain (mathematical analysis)|domain]], as a [[surface integral|hypersurface integral]] of the values of the function on the [[Boundary (topology)|boundary]] of the domain itself. In this respect, it is analogous and generalizes the [[Bochner–Martinelli formula]],<ref>See {{harv|Aizenberg|Yuzhakov|1983|p=38}}, {{harvtxt|Kytmanov|1995|p=9}}, {{harvtxt|Kytmanov|Myslivets|2010|p=20}} and {{harv|Martinelli|1984|pp=152–153}}.</ref> reducing to it when the absolute value of the multiindex order of differentiation is {{math|0}}.<ref>As remarked in {{harv|Kytmanov|1995|p=9}} and {{harv|Kytmanov|Myslivets|2010|p=20}}.</ref> When considered for functions of {{math|''n''&nbsp;<nowiki>=</nowiki>&nbsp;1}} complex variables, it reduces to the ordinary Cauchy formula for the derivative of a holomorphic function:<ref>As remarked by {{harvtxt|Aizenberg|Yuzhakov|1983|p=38}}.</ref> however, when {{math|''n''&nbsp;>&nbsp;1}}, its [[integral kernel]] is not obtainable by simple differentiation of the [[Bochner–Martinelli formula#Bochner–Martinelli kernel|Bochner–Martinelli kernel]].<ref>See the remarks by {{harvtxt|Aizenberg|Yuzhakov|1983|p=38}} and {{harvtxt|Martinelli|1984|loc=p. 153, footnote (1)}}.</ref>

==Historical note==

The Andreotti–Norguet formula was first published in the research announcement {{harv|Andreotti|Norguet|1964|p=780}}:<ref>As correctly stated by {{harvtxt|Aizenberg|Yuzhakov|1983|loc=p. 250, §5}} and {{harvtxt|Kytmanov|1995|p=9}}. {{harvtxt|Martinelli|1984|loc=p. 153, footnote (1)}} cites only the later work {{harv|Andreotti|Norguet|1966}} which, however, contains the full proof of the formula.</ref> however, its full proof was only published later in the paper {{harv|Andreotti|Norguet|1966|pp=207–208}}.<ref>See {{harv|Martinelli|1984|loc=p. 153, footnote (1)}}.</ref> Another, different proof of the formula was given by {{harvtxt|Martinelli|1975}}.<ref>According to {{harvtxt|Aizenberg|Yuzhakov|1983|loc=p. 250, §5}}, {{harvtxt|Kytmanov|1995|p=9}}, {{harvtxt|Kytmanov|Myslivets|2010|p=20}} and {{harvtxt|Martinelli|1984|loc=p. 153, footnote (1)}}, who does not describe his results in this reference, but merely mentions them.</ref> In 1977 and 1978, [[Lev Aizenberg]] gave still another proof and a generalization of the formula based on the [[Cauchy–Fantappiè–Leray formula#Cauchy–Fantappiè–Leray kernel|Cauchy–Fantappiè–Leray kernel]] instead on the [[Bochner–Martinelli formula#Bochner–Martinelli kernel|Bochner–Martinelli kernel]].<ref>See {{harv|Aizenberg|1993|loc=p.289, §13}}, {{harv|Aizenberg|Yuzhakov|1983|loc=p. 250, §5}}, the references cited in those sources and the brief remarks by {{harvtxt|Kytmanov|1995|p=9}} and by {{harvtxt|Kytmanov|Myslivets|2010|p=20}}: each of these works gives Aizenberg's proof.</ref>

==The Andreotti–Norguet integral representation formula==

===Notation===
The notation adopted in the following description of the integral representation formula is the one used by {{harvtxt|Kytmanov|1995|p=9}} and by {{harvtxt|Kytmanov|Myslivets|2010|p=20}}: the notations used in the original works and in other references, though equivalent, are significantly different.<ref>Compare, for example, the original ones by {{harvs|txt|last=Andreotti|last2=Norguet|year=1964|year2=1966|loc=p. 780|loc2=pp. 207–208}} and those used by {{harvtxt|Aizenberg|Yuzhakov|1983|p=38}}, also briefly described in reference {{harv|Aizenberg|1993|p=58}}.</ref> Precisely, it is assumed that

*{{math|''n''&nbsp;>&nbsp;1}} is a fixed [[natural number]], 
*{{math|''ζ'',&nbsp; ''z''&nbsp;∈&nbsp;ℂ{{Sup|''n''}}}} are [[complex vector space|complex vectors]], 
*{{math|''α''&nbsp;<nowiki>=</nowiki>&nbsp;(''α''{{Sub|1}},...,''α''{{Sub|''n''}})&nbsp;∈&nbsp;ℕ{{Sup|''n''}}}} is a [[multiindex]] whose [[Multi-index notation#Definition and basic properties|absolute value]] is {{math|<nowiki>|</nowiki>''α''<nowiki>|</nowiki>}},
*{{math|''D''&nbsp;⊂&nbsp;ℂ{{Sup|''n''}}}} is a bounded domain whose [[Closure (topology)|closure]] is {{math|''{{overline|D}}''}},
*{{math|''A''(''D'')}} is the [[function space]] of functions holomorphic  on the [[Interior (topology)|interior]] of {{math|''D''}} and [[Continuous function|continuous]] on its [[Boundary (topology)|boundary]] {{math|''∂D''}}.
*the iterated Wirtinger derivatives of order {{math|''α''}} of a given complex valued function {{math|''f''&nbsp;∈&nbsp;''A''(''D'')}} are expressed using the following simplified notation:

:<math>\partial^\alpha f = \frac{\partial^{|\alpha|} f}{\partial z_1^{\alpha_1} \cdots \partial z_n^{\alpha_n}}.</math>

===The Andreotti–Norguet kernel===

{{EquationRef|1|Definition 1.}} For every multiindex {{math|''α''}}, the Andreotti–Norguet kernel {{math|''ω''{{Sub|''α''}}&nbsp;(''ζ'',&nbsp;''z'')}} is the following [[differential form]] in {{math|''ζ''}} of bidegree {{math|(''n'',&nbsp;''n''&nbsp;−&nbsp;1)}}:

:<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}(\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},</math>

where {{math|''I''&nbsp;<nowiki>=</nowiki>&nbsp;(1,...,1)&nbsp;∈&nbsp;ℕ{{Sup|''n''}}}} and

:<math> 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}</math>

===The integral formula===

{{EquationRef|2|Theorem 1 (Andreotti and Norguet).}} For every function {{math|''f''&nbsp;∈&nbsp;''A''(''D'')}}, every point {{math|''z''&nbsp;∈&nbsp;''D''}} and every multiindex {{math|''α''}}, the following integral representation formula holds

:<math>\partial^\alpha f(z) = \int_{\partial D} f(\zeta)\omega_\alpha(\zeta,z).</math>

==See also==

*[[Bergman–Weil formula]]

==Notes==

{{reflist|30em}}

==References==
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|archive-url = https://web.archive.org/web/20110927174242/http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=33233
|archive-date = 2011-09-27
|url-status = dead
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{{refend}}

{{DEFAULTSORT:Andreotti-Norguet formula}}
[[Category:Theorems in complex analysis]]
[[Category:Several complex variables]]