View source for Holomorphy, criteria for

''criteria for analyticity''

The natural criteria for holomorphy (analyticity) of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h1201101.png" /> (or continuous) function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h1201102.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h1201103.png" /> of the complex plane are  "infinitesimal"  (cf. [[Analytic function|Analytic function]]), namely: power series expansions, the [[Cauchy–Riemann equations|Cauchy–Riemann equations]], and even the [[Morera theorem|Morera theorem]], since it states that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h1201104.png" /></td> </tr></table>

for all Jordan curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h1201105.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h1201106.png" />, is a necessary and sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h1201107.png" /> being analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h1201108.png" />. The condition (and the usual proofs) depend on the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h1201109.png" /> can be taken to be arbitrarily small.

The first  "non-infinitesimal"  condition is due to M. Agranovsky and R.E. Val'skii (see [[#References|[a2]]] and [[#References|[a6]]] for all relevant references): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011010.png" /> be a piecewise smooth Jordan curve, then a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011011.png" /> continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011012.png" /> is entire (analytic everywhere) if and only if for every transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011014.png" /> it satisfies

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011015.png" /></td> </tr></table>

(Recall that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011016.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011019.png" />.)

A generalization of this theorem and of Morera's theorem which is both local and non-infinitesimal is the following Berenstein–Gay theorem [[#References|[a3]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011020.png" /> be a Jordan polygon contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011022.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011023.png" /> is analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011024.png" /> if and only if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011026.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011027.png" /></td> </tr></table>

This theorem can be extended to several complex variables and other geometries (see [[#References|[a2]]], [[#References|[a5]]], and [[#References|[a6]]] for references).

A different kind of conditions for holomorphy occur when one considers the problem of extending a [[Continuous function|continuous function]] defined on a curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011028.png" /> (or in a real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011029.png" />-manifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011030.png" />) 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|Analytic continuation into a domain of a function given on part of the boundary]]; [[Carleman formulas|Carleman formulas]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011031.png" /> be a subdomain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011032.png" />, bounded by an arc of the unit circle and a smooth simple curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011033.png" /> and assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011034.png" />. Then there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011035.png" />, holomorphic inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011036.png" /> and continuous on its closure, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011037.png" /> if and only if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011038.png" /></td> </tr></table>

A boundary version of the Berenstein–Gay theorem can be proven when regarding the Heisenberg group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011039.png" /> (cf. also [[Nil manifold|Nil manifold]]) as the boundary of the Siegel upper half-space

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011040.png" /></td> </tr></table>

but the boundary values are restricted to be in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011042.png" />, [[#References|[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 [[#References|[a2]]] for references) in the bounded  "version"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011043.png" />, namely the unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011045.png" />, or, more generally, for bounded domains, by essentially considering extensions from the boundary to complex subspaces. An example is the following Globevnik–Stout theorem, [[#References|[a4]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011046.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011047.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011048.png" /> boundary. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011049.png" /> and assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011050.png" /> is such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011051.png" /></td> </tr></table>

for all complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011052.png" />-planes intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011053.png" /> transversally, and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011054.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011055.png" /> with constant coefficients. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011056.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011058.png" />-function, i.e. has an extension as an analytic function to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011059.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Agranovsky,   C. Berenstein,   D.C. Chang,   "Morera theorem for holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120110/h12011060.png" /> functions in the Heisenberg group"  ''J. Reine Angew. Math.'' , '''443'''  (1993)  pp. 49–89</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Berenstein,   D.C. Chang,   D. Pascuas,   L. Zalcman,   "Variations on the theorem of Morera"  ''Contemp. Math.'' , '''137'''  (1992)  pp. 63–78</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Berenstein,   R. Gay,   "Le probléme de Pompeiu local"  ''J. Anal. Math.'' , '''52'''  (1988)  pp. 133–166</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Globevnik,   E.L. Stout,   "Boundary Morera theorems for holomorphic functions of several complex variables"  ''Duke Math. J.'' , '''64'''  (1991)  pp. 571–615</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Zalcman,   "Offbeat integral geometry"  ''Amer. Math. Monthly'' , '''87'''  (1980)  pp. 161–175</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR></table>