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 gave a generalization of the formula by using the Cauchy–Fantappiè–Leray kernel instead of 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

\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 ωα (ζ, z) is the following differential form in ζ of bidegree (n, n − 1):

\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) \, d\bar\zeta^{\alpha+I}[j] \and 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

d\bar\zeta^{\alpha+I}[j] = d\bar\zeta_1^{\alpha_1+1} \and \cdots \and d\bar\zeta_{j-1}^{\alpha_{j+1}+1} \and d\bar\zeta_{j+1}^{\alpha_{j-1}+1} \and \cdots \and 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

\partial^\alpha f(z) = \int_{\partial D} f(\zeta)\omega_\alpha(\zeta,z).

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 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), Kytmanov & Myslivets (2010, p. 20): each of these works report 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 the .

References