Holomorphy, criteria for
criteria for analyticity
The natural criteria for holomorphy (analyticity) of a (or continuous) function
in a domain
of the complex plane are "infinitesimal" (cf. Analytic function), namely: power series expansions, the Cauchy–Riemann equations, and even the Morera theorem, since it states that
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for all Jordan curves such that
, is a necessary and sufficient condition for
being analytic in
. The condition (and the usual proofs) depend on the fact that
can be taken to be arbitrarily small.
The first "non-infinitesimal" condition is due to M. Agranovsky and R.E. Val'skii (see [a2] and [a6] for all relevant references): Let be a piecewise smooth Jordan curve, then a function
continuous in
is entire (analytic everywhere) if and only if for every transformation
and
it satisfies
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(Recall that means that
,
,
.)
A generalization of this theorem and of Morera's theorem which is both local and non-infinitesimal is the following Berenstein–Gay theorem [a3].
Let be a Jordan polygon contained in
and
; then
is analytic in
if and only if for any
such that
,
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This theorem can be extended to several complex variables and other geometries (see [a2], [a5], and [a6] for references).
A different kind of conditions for holomorphy occur when one considers the problem of extending a continuous function defined on a curve in (or in a real
-manifold in
) to an analytic function defined in a domain that contains the curve (or hypersurface) on its boundary. This is sometimes called a CR extension. An example of this type, generalizing the moment conditions of the Berenstein–Gay theorem, appears in the work of L. Aizenberg and collaborators (cf. also Analytic continuation into a domain of a function given on part of the boundary; Carleman formulas): Let
be a subdomain of
, bounded by an arc of the unit circle and a smooth simple curve
and assume that
. Then there is a function
, holomorphic inside
and continuous on its closure, such that
if and only if
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A boundary version of the Berenstein–Gay theorem can be proven when regarding the Heisenberg group (cf. also Nil manifold) as the boundary of the Siegel upper half-space
![]() |
but the boundary values are restricted to be in ,
, [a1]. Related analytic extension theorems from continuous boundary values have been proven by J. Globevnik and E.L. Stout, E. Grinberg, W. Rudin, and others (see [a2] for references) in the bounded "version" of
, namely the unit ball
of
, or, more generally, for bounded domains, by essentially considering extensions from the boundary to complex subspaces. An example is the following Globevnik–Stout theorem, [a4].
Let be a bounded domain in
with
boundary. Let
and assume
is such that
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for all complex -planes intersecting
transversally, and all
-forms
with constant coefficients. Then
is a
-function, i.e. has an extension as an analytic function to
.
References
[a1] | M. Agranovsky, C. Berenstein, D.C. Chang, "Morera theorem for holomorphic ![]() |
[a2] | C. Berenstein, D.C. Chang, D. Pascuas, L. Zalcman, "Variations on the theorem of Morera" Contemp. Math. , 137 (1992) pp. 63–78 |
[a3] | C. Berenstein, R. Gay, "Le probléme de Pompeiu local" J. Anal. Math. , 52 (1988) pp. 133–166 |
[a4] | J. Globevnik, E.L. Stout, "Boundary Morera theorems for holomorphic functions of several complex variables" Duke Math. J. , 64 (1991) pp. 571–615 |
[a5] | L. Zalcman, "Offbeat integral geometry" Amer. Math. Monthly , 87 (1980) pp. 161–175 |
[a6] | L. Zalcman, "A bibliographic survey of the Pompeiu problem" B. Fuglede (ed.) et al. (ed.) , Approximation by Solutions of Partial Differential equations , Kluwer Acad. Publ. (1992) pp. 185–194 |
Holomorphy, criteria for. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Holomorphy,_criteria_for&oldid=31195