Andreotti–Norguet formula

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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):[7] however, its full proof was only published later in the paper (Andreotti Norguet).[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

Notation

The notation adopted in the following description of the integral representation formula is the one used by (Kytmanov 1995) and by (Kytmanov Myslivets): 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,
  • [math]\displaystyle{ \zeta, z \in \Complex^n }[/math] are complex vectors,
  • [math]\displaystyle{ \alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{N}^n }[/math] is a multiindex whose absolute value is |α|,
  • [math]\displaystyle{ D \subset \Complex^n }[/math] 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: [math]\displaystyle{ \partial^\alpha f = \frac{\partial^{|\alpha|} f}{\partial z_1^{\alpha_1} \cdots \partial z_n^{\alpha_n}}. }[/math]

The Andreotti–Norguet kernel

Definition 1. For every multiindex α, the Andreotti–Norguet kernel ωα (ζ, z) is the following differential form in ζ of bidegree (n, n − 1): [math]\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}, }[/math] where [math]\displaystyle{ I = (1, \dots, 1) \in \N^n }[/math] and [math]\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} }[/math]

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

See also

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), (Kytmanov 1995), (Kytmanov Myslivets) and (Martinelli 1984).
  4. ↑ As remarked in (Kytmanov 1995) and (Kytmanov Myslivets).
  5. ↑ As remarked by (Aizenberg Yuzhakov).
  6. ↑ See the remarks by (Aizenberg Yuzhakov) and (Martinelli 1984).
  7. ↑ As correctly stated by (Aizenberg Yuzhakov) and (Kytmanov 1995). (Martinelli 1984) cites only the later work (Andreotti Norguet) which, however, contains the full proof of the formula.
  8. ↑ See (Martinelli 1984).
  9. ↑ According to (Aizenberg Yuzhakov), (Kytmanov 1995), (Kytmanov Myslivets) and (Martinelli 1984), who does not describe his results in this reference, but merely mentions them.
  10. ↑ See (Aizenberg 1993), (Aizenberg Yuzhakov), the references cited in those sources and the brief remarks by (Kytmanov 1995) and by (Kytmanov Myslivets): 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), also briefly described in reference (Aizenberg 1993).

References



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