Andreotti–Norguet formula

The Andreotti–Norguet formula, first introduced by Aldo Andreotti and François Norguet (1964, 1966),^[1] is a higher–dimensional analogue of Cauchy integral formula for expressing the derivatives of a holomorphic function. Precisely, this formula express the value of the partial derivative of any multiindex order of a holomorphic function of several variables,^[2] in any interior point of a given bounded domain, as a hypersurface integral of the values of the function on the boundary of the domain itself. In this respect, it is analogous and generalizes the Bochner–Martinelli formula,^[3] reducing to it when the absolute value of the multiindex order of differentiation is 0.^[4] When considered for functions of n = 1 complex variables, it reduces to the ordinary Cauchy formula for the derivative of a holomorphic function:^[5] however, when n > 1, its integral kernel is not obtainable by simple differentiation of the Bochner–Martinelli kernel.^[6]

Historical note

The Andreotti–Norguet formula was first published in the research announcement (Andreotti & Norguet 1964, p. 780):^[7] however, its full proof was only published later in the paper (Andreotti & Norguet 1966, pp. 207–208).^[8] Another, different proof of the formula was given by Martinelli (1975).^[9] In 1977 and 1978, Lev Aizenberg gave still another proof and a generalization of the formula based on the Cauchy–Fantappiè–Leray kernel instead on the Bochner–Martinelli kernel.^[10]

The Andreotti–Norguet integral representation formula

The notation adopted in the following description of the integral representation formula is the one used by Kytmanov (1995, p. 9) and by Kytmanov & Myslivets (2010, p. 20): the notations used in the original works and in other references, though equivalent, are significantly different.^[11] Precisely, it is assumed that

• n > 1 is a fixed natural number,
• ζ,  z ∈ ℂⁿ are complex vectors,
• α = (α₁,...,α_n) ∈ ℕⁿ is a multiindex whose absolute value is |α|,
• D ⊂ ℂⁿ is a bounded domain whose closure is D,
• A(D) is the function space of functions holomorphic on the interior of D and continuous on its boundary ∂D.
• the iterated Wirtinger derivatives of order α of a given complex valued function f ∈ A(D) are expressed using the following simplified notation:

${\displaystyle \partial ^{\alpha }f={\frac {\partial ^{|\alpha |}f}{\partial z_{1}^{\alpha _{1}}\cdots \partial z_{n}^{\alpha _{n}}}}.}$

The Andreotti–Norguet kernel

Definition 1. For every multiindex α, the Andreotti–Norguet kernel ω_α (ζ, z) is the following differential form in ζ of bidegree (n, n − 1):

${\displaystyle \omega _{\alpha }(\zeta ,z)={\frac {(n-1)!\alpha _{1}!\cdots \alpha _{n}!}{(2\pi i)^{n}}}\sum _{j=1}^{n}{\frac {(-1)^{j-1}({\bar {\zeta }}_{j}-{\overline {z}}_{j})^{\alpha _{j}+1}\,d{\bar {\zeta }}^{\alpha +I}[j]\land d\zeta }{\left(|z_{1}-\zeta _{1}|^{2(\alpha _{1}+1)}+\cdots +|z_{n}-\zeta _{n}|^{2(\alpha _{n}+1)}\right)^{n}}},}$

where I = (1,...,1) ∈ ℕⁿ and

${\displaystyle d{\bar {\zeta }}^{\alpha +I}[j]=d{\bar {\zeta }}_{1}^{\alpha _{1}+1}\land \cdots \land d{\bar {\zeta }}_{j-1}^{\alpha _{j+1}+1}\land d{\bar {\zeta }}_{j+1}^{\alpha _{j-1}+1}\land \cdots \land d{\bar {\zeta }}_{n}^{\alpha _{n}+1}}$

The integral formula

Theorem 1 (Andreotti and Norguet). For every function f ∈ A(D), every point z ∈ D and every multiindex α, the following integral representation formula holds

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

See also

• Bergman–Weil formula

Notes

1. For a brief historical sketch, see the "historical section" of the present entry.
2. Partial derivatives of a holomorphic function of several complex variables are defined as partial derivatives respect to its complex arguments, i.e. as Wirtinger derivatives.
3. See (Aizenberg & Yuzhakov 1983, p. 38), Kytmanov (1995, p. 9), Kytmanov & Myslivets (2010, p. 20) and (Martinelli 1984, pp. 152–153).
4. As remarked in (Kytmanov 1995, p. 9) and (Kytmanov & Myslivets 2010, p. 20).
5. As remarked by Aizenberg & Yuzhakov (1983, p. 38).
6. See the remarks by Aizenberg & Yuzhakov (1983, p. 38) and Martinelli (1984, p. 153, footnote (1)).
7. As correctly stated by Aizenberg & Yuzhakov (1983, p. 250, §5) and Kytmanov (1995, p. 9). Martinelli (1984, p. 153, footnote (1)) cites only the later work (Andreotti & Norguet 1966) which, however, contains the full proof of the formula.
8. See (Martinelli 1984, p. 153, footnote (1)).
9. According to Aizenberg & Yuzhakov (1983, p. 250, §5), Kytmanov (1995, p. 9), Kytmanov & Myslivets (2010, p. 20) and Martinelli (1984, p. 153, footnote (1)), who does not describe his results in this reference, but merely mentions them.
10. See (Aizenberg 1993, p.289, §13), (Aizenberg & Yuzhakov 1983, p. 250, §5), the references cited in those sources and the brief remarks by Kytmanov (1995, p. 9) and by Kytmanov & Myslivets (2010, p. 20): each of these works gives Aizenberg's proof.
11. Compare, for example, the original ones by Andreotti and Norguet (1964, p. 780, 1966, pp. 207–208) and those used by Aizenberg & Yuzhakov (1983, p. 38), also briefly described in reference (Aizenberg 1993, p. 58).

References

• Aizenberg, Lev (1993) [1990], Carleman's Formulas in Complex Analysis. Theory and applications, Mathematics and Its Applications, 244 (2nd ed.), Dordrecht–Boston–London: Kluwer Academic Publishers, pp. xx+299, doi:10.1007/978-94-011-1596-4, ISBN 0-7923-2121-9, MR 1256735, Zbl 0783.32002, revised translation of the 1990 Russian original.
• Aizenberg, L. A.; Yuzhakov, A. P. (1983) [1979], 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, MR 0735793, Zbl 0537.32002.
• Andreotti, Aldo; Norguet, François (20 January 1964), "Problème de Levi pour les classes de cohomologie", Comptes rendus hebdomadaires des séances de l'Académie des Sciences (in French), 258 (Première partie): 778–781, MR 0159960, Zbl 0124.38803.
• Andreotti, Aldo; Norguet, François (1966), "Problème de Levi et convexité holomorphe pour les classes de cohomologie", Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Serie III, (in French), 20 (2): 197–241, MR 199439, Zbl 0154.3350.
• Berenstein, Carlos A.; Gay, Roger; Vidras, Alekos; Yger, Alain (1993), Residue currents and Bezout identities, Progress in Mathematics, 114, Basel–Berlin–Boston: Birkhäuser Verlag, pp. xi+158, doi:10.1007/978-3-0348-8560-7, ISBN 3-7643-2945-9, MR 1249478, Zbl 0802.32001 ISBN 0-8176-2945-9, ISBN 978-3-0348-8560-7.
• Kytmanov, Alexander M. (1995) [1992], The Bochner–Martinelli integral and its applications, Birkhäuser Verlag, pp. xii+305, 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.
• 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 06482036, ISBN 978-3-319-21659-1 (ebook).
• Martinelli, Enzo (1975), "Sopra una formula di Andreotti–Norguet", Bollettino dell'Unione Matematica Italiana, IV Serie (in Italian), 11 (3, Supplemento): 455–457, MR 0390270, Zbl 0317.32006. Collection of articles dedicated to Giovanni Sansone on the occasion of his eighty-fifth birthday.
• Martinelli, Enzo (1984), 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 (in Italian), 67, Rome: Accademia Nazionale dei Lincei, pp. 236+II. "Elementary introduction to the theory of functions of complex variables with particular regard to integral representations" (English translation of the title reads) is a textbook, published by the Accademia Nazionale dei Lincei, taken from the notes from a course held by Martinelli when he was in charge to the academy as a "Professore Linceo".