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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

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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

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Notation

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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,
  • are complex vectors,
  • is a multiindex whose absolute value is |α|,
  • 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 fA(D) are expressed using the following simplified notation:

The Andreotti–Norguet kernel

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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): where and

The integral formula

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Theorem 1 (Andreotti and Norguet). For every function fA(D), every point zD and every multiindex α, the following integral representation formula holds

See also

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Notes

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  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

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