Bochner–Martinelli formula
In mathematics, the Bochner–Martinelli formula is a generalization of the
).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 the Bochner–Martinelli kernel ω(ζ,z) is a differential form in ζ of bidegree (n,n−1) defined by
(where the term dζj is omitted).
Suppose that f is a continuously differentiable function on the closure of a domain D in n with piecewise smooth boundary ∂D. Then the Bochner–Martinelli formula states that if z is in the domain D then
In particular if f is holomorphic the second term vanishes, so
See also
Notes
- ^ 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).
- ^ Bochner refers to his claim in (Bochner 1943, p. 652, footnote 1).
References
- Zbl 0537.32002.
- Zbl 0060.24206.
- Zbl 0038.23701.
- Chirka, E.M. (2001) [1994], "Bochner–Martinelli representation formula", Encyclopedia of Mathematics, EMS Press
- Zbl 1087.32001.
- Zbl 0834.32001.
- ISBN 978-5-7638-1990-8, archived from the originalon 2014-03-23.
- 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], Bochner-Martinelli formulais 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], 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, 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, Zbl 0599.32002. In this article, Martinelli gives another form to the Martinelli–Bochner formula.