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 EnzoMartinelli  ( 1938 ) and SalomonBochner  ( 1943 ).

History

Formula (53) of the present paper and a proof of theorem 5 based on it have just been published by Enzo Martinelli (...). ^[1] 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 k variables with some applications.

— Salomon Bochner, (Bochner 1943, p. 652, footnote 1).

However this author's claim in loc. cit. footnote 1, ^[2] that he might have been familiar with the general shape of the formula before Martinelli, was wholly unjustified and is hereby being retracted.

— Salomon Bochner, (Bochner 1947, p. 15, footnote *).

Bochner–Martinelli kernel

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

${\displaystyle \omega (\zeta ,z)={\frac {(n-1)!}{(2\pi i)^{n}}}{\frac {1}{|z-\zeta |^{2n}}}\sum _{1\leq j\leq n}({\overline {\zeta }}_{j}-{\overline {z}}_{j})\,d{\overline {\zeta }}_{1}\land d\zeta _{1}\land \cdots \land d\zeta _{j}\land \cdots \land d{\overline {\zeta }}_{n}\land d\zeta _{n}}$

(where the term dζ_j is omitted).

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

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

In particular if f is holomorphic the second term vanishes, so

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

See also

• Bergman–Weil formula

Notes

1. Bochner refers explicitly to the article (Martinelli 1942–1943), apparently being not aware of the earlier one (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 (Martinelli 1942–1943, p. 340, footnote 2).
2. Bochner refers to his claim in (Bochner 1943, p. 652, footnote 1).

References

• Aizenberg, L. A.; Yuzhakov, A. P. (1983) [1979], Integral Representations and Residues in Multidimensional Complex Analysis, Translations of Mathematical Monographs, vol. 58, Providence R.I.: American Mathematical Society, pp. x+283, ISBN   0-8218-4511-X, MR   0735793, Zbl   0537.32002 .
• Bochner, Salomon (1943), "Analytic and meromorphic continuation by means of Green's formula", Annals of Mathematics , Second Series, 44 (4): 652–673, doi:10.2307/1969103, ISSN   0003-486X, JSTOR   1969103, MR   0009206, Zbl   0060.24206 .
• Bochner, Salomon (1947), "On compact complex manifolds", The Journal of the Indian Mathematical Society , New Series, 11: 1–21, MR   0023919, Zbl   0038.23701 .
• Chirka, E.M. (2001) [1994], "Bochner–Martinelli representation formula", Encyclopedia of Mathematics , EMS Press
• Krantz, Steven G. (2001) [1992], Function theory of several complex variables (reprint of 2nd ed.), Providence, R.I.: AMS Chelsea Publishing, pp. xvi+564, doi:10.1090/chel/340, ISBN   978-0-8218-2724-6, MR   1846625, Zbl   1087.32001 .
• Kytmanov, Alexander M. (1995) [1992], The Bochner-Martinelli integral and its applications, Birkhäuser Verlag, pp. xii+305, doi:10.1007/978-3-0348-9094-6, ISBN   978-3-7643-5240-0, MR   1409816, Zbl   0834.32001 .
• Kytmanov, Alexander M.; Myslivets, Simona G. (2010), Интегральные представления и их приложения в многомерном комплексном анализе [Integral representations and their application in multidimensional complex analysis], Красноярск: СФУ, p. 389, ISBN   978-5-7638-1990-8, archived from the original on 2014-03-23.
• Kytmanov, Alexander M.; Myslivets, Simona G. (2015), Multidimensional integral representations. Problems of analytic continuation, Cham–Heidelberg–New York–Dordrecht–London: Springer Verlag, pp. xiii+225, doi:10.1007/978-3-319-21659-1, ISBN   978-3-319-21658-4, MR   3381727, Zbl   1341.32001 , ISBN   978-3-319-21659-1 (ebook).
• Martinelli, Enzo (1938), "Alcuni teoremi integrali per le funzioni analitiche di più variabili complesse" [Some integral theorems for analytic functions of several complex variables], Atti della Reale Accademia d'Italia. Memorie della Classe di Scienze Fisiche, Matematiche e Naturali (in Italian), 9 (7): 269–283, JFM   64.0322.04, Zbl   0022.24002 . The first paper where the now called Bochner-Martinelli formula is introduced and proved.
• Martinelli, Enzo (1942–1943), "Sopra una dimostrazione di R. Fueter per un teorema di Hartogs" [On a proof of R. Fueter of a theorem of Hartogs], Commentarii Mathematici Helvetici (in Italian), 15 (1): 340–349, doi:10.1007/bf02565649, MR   0010729, S2CID   119960691, Zbl   0028.15201, archived from the original on 2011-10-02, retrieved 2020-07-04. Available at the SEALS Portal Archived 2012-11-10 at the Wayback Machine . In this paper Martinelli gives a proof of Hartogs' extension theorem by using the Bochner-Martinelli formula.
• Martinelli, Enzo (1984), Introduzione elementare alla teoria delle funzioni di variabili complesse con particolare riguardo alle rappresentazioni integrali [Elementary introduction to the theory of functions of complex variables with particular regard to integral representations], Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni (in Italian), vol. 67, Rome: Accademia Nazionale dei Lincei, pp. 236+II, archived from the original on 2011-09-27, retrieved 2011-01-03. The notes form a course, published by the Accademia Nazionale dei Lincei, held by Martinelli during his stay at the Accademia as "Professore Linceo".
• Martinelli, Enzo (1984b), "Qualche riflessione sulla rappresentazione integrale di massima dimensione per le funzioni di più variabili complesse" [Some reflections on the integral representation of maximal dimension for functions of several complex variables], Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali, Series VIII (in Italian), 76 (4): 235–242, MR   0863486, Zbl   0599.32002 . In this article, Martinelli gives another form to the Martinelli–Bochner formula.