Bochner–Martinelli formula

{{Quote

|text= Formula (53) of the present paper and a proof of theorem 5 based on it have just been published by Enzo Martinelli (...).Bochner refers explicitly to the article {{harv|Martinelli|1942–1943}}, apparently being not aware of the earlier one {{harv|Martinelli|1938}}, which actually contains Martinelli's proof of the formula. However, the earlier article is explicitly cited in the later one, as it can be seen from {{harv|Martinelli|1942–1943|loc=p. 340, footnote 2}}. The present author may be permitted to state that these results have been presented by him in a Princeton graduate course in Winter 1940/1941 and were subsequently incorporated, in a Princeton doctorate thesis (June 1941) by Donald C. May, entitled: An integral formula for analytic functions of {{math|k}} variables with some applications.

|sign=Salomon Bochner

|source={{harv|Bochner|1943|loc=p. 652, footnote 1}}.

}}

{{Quote

|text= However this author's claim in loc. cit. footnote 1,Bochner refers to his claim in {{harv|Bochner|1943|loc=p. 652, footnote 1}}. that he might have been familiar with the general shape of the formula before Martinelli, was wholly unjustified and is hereby being retracted.

|sign=Salomon Bochner

|source={{harv|Bochner|1947|loc=p. 15, footnote *}}.

}}

For {{math|ζ}}, {{math|z}} in $\backslash C^n$ the Bochner–Martinelli kernel {{math|ω(ζ,z)}} is a differential form in {{math|ζ}} of bidegree {{math|(n,n−1)}} defined by

:$\backslash omega(\backslash zeta,z)\; =\; \backslash frac\{(n-1)!\}\{(2\backslash pi\; i)^n\}\backslash frac\{1\}\{|z-\backslash zeta|^\{2n\}\}$

\sum_{1\le j\le n}(\overline\zeta_j-\overline z_j) \, d\overline\zeta_1 \land d\zeta_1 \land \cdots \land d\zeta_j \land \cdots \land d\overline\zeta_n \land d\zeta_n

(where the term {{math|d{{overline|ζ}}_j}} is omitted).

Suppose that {{math|f}} is a continuously differentiable function on the closure of a domain {{math|D}} in $\backslash mathbb\{C\}$ⁿ with piecewise smooth boundary {{math|∂D}}. Then the Bochner–Martinelli formula states that if {{math|z}} is in the domain {{math|D}} then

:$\backslash displaystyle\; f(z)\; =\; \backslash int\_\{\backslash partial\; D\}f(\backslash zeta)\backslash omega(\backslash zeta,\; z)\; -\; \backslash int\_D\backslash overline\backslash partial\; f(\backslash zeta)\backslash land\backslash omega(\backslash zeta,z).$

In particular if {{math|f}} is holomorphic the second term vanishes, so

:$\backslash displaystyle\; f(z)\; =\; \backslash int\_\{\backslash partial\; D\}f(\backslash zeta)\backslash omega(\backslash zeta,\; z).$

• Bergman–Weil formula

{{Reflist|30em}}

{{Refbegin}}

• {{Citation

|last1 = Aizenberg

|first1 = L. A.

|author-link = Lev Aizenberg

|last2 = Yuzhakov

|first2 = A. P.

|author2-link = Aleksandr Yuzhakov

|title = Integral Representations and Residues in Multidimensional Complex Analysis

|url = https://books.google.com/books?id=2ZWsf6ufee8C

|place = Providence R.I.

|series = Translations of Mathematical Monographs

|volume = 58

|publisher = American Mathematical Society

|pages = x+283

|year = 1983

|orig-year = 1979

|isbn = 0-8218-4511-X

|mr = 0735793

|zbl = 0537.32002

}}.

• {{Citation

|last1=Bochner

|first1=Salomon

|author1-link=Salomon Bochner

|title=Analytic and meromorphic continuation by means of Green's formula

|jstor=1969103

|mr=0009206

|zbl = 0060.24206

|series=Second Series

|year=1943

|journal=Annals of Mathematics

|issn=0003-486X

|doi=10.2307/1969103

|volume=44

|issue=4

|pages=652–673

}}.

• {{Citation

|last1 = Bochner

|first1 = Salomon

|author1-link = Salomon Bochner

|title = On compact complex manifolds

|journal = The Journal of the Indian Mathematical Society

|series = New Series

|volume = 11

|pages = 1–21

|year = 1947

|url =

|mr = 0023919

|zbl = 0038.23701

}}.

• {{eom|id=b/b016720|first=E.M.|last= Chirka|title=Bochner–Martinelli representation formula}}
• {{Citation

|last=Krantz

|first=Steven G.

|author-link=Steven G. Krantz

|title=Function theory of several complex variables

|url=https://books.google.com/books?isbn=9780821827246

|publisher=AMS Chelsea Publishing

|year=2001

|orig-year=1992

|edition=reprint of 2nd

|pages=xvi+564

|place=Providence, R.I.

|isbn=978-0-8218-2724-6

|mr=1846625

|zbl=1087.32001

|doi=10.1090/chel/340

}}.

• {{Citation

|last = Kytmanov

|first = Alexander M.

|author-link = Alexander Kytmanov

|title = The Bochner-Martinelli integral and its applications

|orig-year = 1992

|url = https://books.google.com/books?isbn=376435240X

|publisher = Birkhäuser Verlag

|pages = xii+305

|year = 1995

|isbn = 978-3-7643-5240-0

|mr = 1409816

|zbl = 0834.32001

|doi = 10.1007/978-3-0348-9094-6

}}.

• {{Citation

|last1 = Kytmanov

|first1 = Alexander M.

|author-link = Alexander Kytmanov

|last2 = Myslivets

|first2 = Simona G.

|title = Интегральные представления и их приложения в многомерном комплексном анализе

|trans-title = Integral representations and their application in multidimensional complex analysis

|url = http://www.eastview.com/russian/books/product.asp?SKU=930345B&f_locale=_CYR&active_tab=1

|place = Красноярск

|publisher = СФУ

|pages = 389

|year = 2010

|isbn = 978-5-7638-1990-8

|url-status = dead

|archiveurl = https://web.archive.org/web/20140323020317/http://www.eastview.com/russian/books/product.asp?SKU=930345B&f_locale=_CYR&active_tab=1

|archivedate = 2014-03-23

}}.

• {{Citation

|last1 = Kytmanov

|first1 = Alexander M.

|author-link = Alexander Kytmanov

|last2 = Myslivets

|first2 = Simona G.

|author2-link =

|title = Multidimensional integral representations. Problems of analytic continuation

|url = https://books.google.com/books?id=jpWKCgAAQBAJ

|place = Cham–Heidelberg–New York–Dordrecht–London

|publisher = Springer Verlag

|pages = xiii+225

|year = 2015

|isbn = 978-3-319-21658-4

|doi = 10.1007/978-3-319-21659-1

|mr = 3381727

|zbl = 1341.32001

}}, {{ISBN|978-3-319-21659-1}} (ebook).

• {{Citation

|last=Martinelli

|first=Enzo

|author-link=

|title=Alcuni teoremi integrali per le funzioni analitiche di più variabili complesse

|trans-title=Some integral theorems for analytic functions of several complex variables

|language = Italian

|year=1938

|journal=Atti della Reale Accademia d'Italia. Memorie della Classe di Scienze Fisiche, Matematiche e Naturali

|issue=7

|volume=9

|pages=269–283

|id=

|jfm= 64.0322.04

|zbl = 0022.24002

}}. The first paper where the now called Bochner-Martinelli formula is introduced and proved.

• {{Citation

|last = Martinelli

|first = Enzo

|title = Sopra una dimostrazione di R. Fueter per un teorema di Hartogs

|trans-title = On a proof of R. Fueter of a theorem of Hartogs

|language = Italian

|journal = Commentarii Mathematici Helvetici

|volume = 15

|issue = 1

|pages = 340–349

|year = 1942–1943

|url = http://retro.seals.ch/digbib/en/view?rid=comahe-002:1942–1943:15::26

|doi = 10.1007/bf02565649

|mr = 0010729

|zbl = 0028.15201

|s2cid = 119960691

|url-status = dead

|archiveurl = https://web.archive.org/web/20111002072948/http://retro.seals.ch/digbib/en/view?rid=comahe-002%3A1942%E2%80%931943%3A15%3A%3A26

|archivedate = 2011-10-02

|access-date = 2020-07-04

}}. Available at the [http://retro.seals.ch/digbib/home SEALS Portal] {{Webarchive|url=https://web.archive.org/web/20121110040541/http://retro.seals.ch/digbib/home |date=2012-11-10 }}. In this paper Martinelli gives a proof of Hartogs' extension theorem by using the Bochner-Martinelli formula.

• {{Citation

|last = Martinelli

|first = Enzo

|author-link =

|title = Introduzione elementare alla teoria delle funzioni di variabili complesse con particolare riguardo alle rappresentazioni integrali

|trans-title = Elementary introduction to the theory of functions of complex variables with particular regard to integral representations

|language = Italian

|place = Rome

|publisher = Accademia Nazionale dei Lincei

|year = 1984

|series = Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni

|volume = 67

|pages = 236+II

|url = http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=33233

|doi =

|id =

|isbn =

|access-date = 2011-01-03

|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

}}. The notes form a course, published by the Accademia Nazionale dei Lincei, held by Martinelli during his stay at the Accademia as "Professore Linceo".

• {{Citation

|last = Martinelli

|first = Enzo

|title = Qualche riflessione sulla rappresentazione integrale di massima dimensione per le funzioni di più variabili complesse

|trans-title = Some reflections on the integral representation of maximal dimension for functions of several complex variables

|language = Italian

|journal = Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali

|series = Series VIII

|url = http://www.bdim.eu/item?fmt=pdf&id=RLIN_1984_8_76_4_235_0

|volume = 76

|issue = 4

|pages = 235–242

|year = 1984b

|mr = 0863486

|zbl = 0599.32002

}}. In this article, Martinelli gives another form to the Martinelli–Bochner formula.

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{{DEFAULTSORT:Bochner-Martinelli formula}}

Category:Theorems in complex analysis

Category:Several complex variables