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 Enzo Martinelli (1938) and Salomon Bochner (1943).

History

Formula (53) of the present paper and a proof of theorem 5 based on it have just been published by Enzo Martinelli (...). 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).

However this author's claim in loc. cit. 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.

—Salomon Bochner, (Bochner 1947).

Bochner–Martinelli kernel

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

[math]\displaystyle{ \omega(\zeta,z) = \frac{(n-1)!}{(2\pi i)^n}\frac{1}{|z-\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 }[/math]

(where the term dζ_j is omitted).

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

[math]\displaystyle{ \displaystyle f(z) = \int_{\partial D}f(\zeta)\omega(\zeta, z) - \int_D\overline\partial f(\zeta)\land\omega(\zeta,z). }[/math]

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

[math]\displaystyle{ \displaystyle f(z) = \int_{\partial D}f(\zeta)\omega(\zeta, z). }[/math]

See also

• Bergman–Weil formula

References

• Aizenberg, L. A.; Yuzhakov, A. P. (1983), Integral Representations and Residues in Multidimensional Complex Analysis, Translations of Mathematical Monographs, 58, Providence R.I.: American Mathematical Society, pp. x+283, ISBN 0-8218-4511-X, https://books.google.com/books?id=2ZWsf6ufee8C .
• 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 .
• Bochner, Salomon (1947), "On compact complex manifolds", The Journal of the Indian Mathematical Society, New Series 11: 1–21 .
• Hazewinkel, Michiel, ed. (2001), "Bochner–Martinelli representation formula", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=b/b016720 
• Krantz, Steven G. (2001), 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, https://books.google.com/books?isbn=9780821827246 .
• Kytmanov, Alexander M. (1995), 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, https://books.google.com/books?isbn=376435240X .
• Kytmanov, Alexander M.; Myslivets, Simona G. (2010), Интегральные представления и их приложения в многомерном комплексном анализе, Красноярск: СФУ, pp. 389, ISBN 978-5-7638-1990-8, archived from the original on 2014-03-23, https://web.archive.org/web/20140323020317/http://www.eastview.com/russian/books/product.asp?SKU=930345B&f_locale=_CYR&active_tab=1 .
• 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, https://books.google.com/books?id=jpWKCgAAQBAJ , ISBN:978-3-319-21659-1 (ebook).
• Martinelli, Enzo (1938), "Alcuni teoremi integrali per le funzioni analitiche di più variabili complesse" (in Italian), Atti della Reale Accademia d'Italia. Memorie della Classe di Scienze Fisiche, Matematiche e Naturali 9 (7): 269–283 . 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" (in Italian), Commentarii Mathematici Helvetici 15 (1): 340–349, doi:10.1007/bf02565649, archived from the original on 2011-10-02, https://web.archive.org/web/20111002072948/http://retro.seals.ch/digbib/en/view?rid=comahe-002%3A1942%E2%80%931943%3A15%3A%3A26, retrieved 2020-07-04 . Available at the SEALS Portal . In this paper Martinelli gives a proof of Hartogs' extension theorem by using the Bochner-Martinelli formula.
• Martinelli, Enzo (1984) (in Italian), Introduzione elementare alla teoria delle funzioni di variabili complesse con particolare riguardo alle rappresentazioni integrali, Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni, 67, Rome: Accademia Nazionale dei Lincei, pp. 236+II, http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=33233, 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" (in Italian), Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali, Series VIII 76 (4): 235–242, http://www.bdim.eu/item?fmt=pdf&id=RLIN_1984_8_76_4_235_0 . In this article, Martinelli gives another form to the Martinelli–Bochner formula.