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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), 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, 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, reducing to it when the absolute value of the multiindex order of differentiation is 0. When considered for functions of n = 1 complex variables, it reduces to the ordinary Cauchy formula for the derivative of a holomorphic function: however, when n > 1, its integral kernel is not obtainable by simple differentiation of the Bochner–Martinelli kernel.

Historical note

The Andreotti–Norguet formula was first published in the research announcement (Andreotti & Norguet 1964, p. 780): however, its full proof was only published later in the paper (Andreotti & Norguet 1966, pp. 207–208). Another, different proof of the formula was given by Martinelli (1975). 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.

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, 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. Precisely, it is assumed that

  • n > 1 is a fixed natural number,
  • ζ , z ∈ C n {\displaystyle \zeta ,z\in \mathbb {C} ^{n}} are complex vectors,
  • α = ( α 1 , … , α n ) ∈ N n {\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})\in \mathbb {N} ^{n}} is a multiindex whose absolute value is |α|,
  • D ⊂ C n {\displaystyle D\subset \mathbb {C} ^{n}} 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: ∂ α f = ∂ | α | f ∂ z 1 α 1 ⋯ ∂ z n α n . {\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 ωα (ζ, z) is the following differential form in ζ of bidegree (n, n − 1): ω α ( ζ , z ) = ( n − 1 ) ! α 1 ! ⋯ α n ! ( 2 π i ) n ∑ j = 1 n ( − 1 ) j − 1 ( ζ ¯ j − z ¯ j ) α j + 1 d ζ ¯ α + I [ j ] ∧ d ζ ( | z 1 − ζ 1 | 2 ( α 1 + 1 ) + ⋯ + | z n − ζ n | 2 ( α n + 1 ) ) n , {\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}\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 ) ∈ N n {\displaystyle I=(1,\dots ,1)\in \mathbb {N} ^{n}} and d ζ ¯ α + I [ j ] = d ζ ¯ 1 α 1 + 1 ∧ ⋯ ∧ d ζ ¯ j − 1 α j + 1 + 1 ∧ d ζ ¯ j + 1 α j − 1 + 1 ∧ ⋯ ∧ d ζ ¯ n α n + 1 {\displaystyle d{\bar {\zeta }}^{\alpha +I}=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 ∂ α f ( z ) = ∫ ∂ D f ( ζ ) ω α ( ζ , z ) . {\displaystyle \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) 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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