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] 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 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 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):
( ζ , z ) =
ζ ¯
ζ ¯
[ j ] ∧ d ζ
ζ
ζ
ζ ¯
ζ ¯
ζ ¯
ζ ¯
ζ ¯
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