View source for Hartogs-Laurent series

A series

<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/h046/h046640/h0466401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h0466402.png" /> and where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h0466403.png" /> are functions holomorphic in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h0466404.png" /> which is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h0466405.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h0466406.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h0466407.png" />, the series (*) is known as a Hartogs series. Any holomorphic function in a [[Hartogs domain|Hartogs domain]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h0466408.png" /> of the type

<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/h046/h046640/h0466409.png" /></td> </tr></table>

can be expanded into a Hartogs–Laurent series which converges absolutely and uniformly inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h04664010.png" />. In complete Hartogs domains this will be the expansion into a Hartogs series. The domains of convergence of Hartogs–Laurent series are domains of the same kind with special <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h04664011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h04664012.png" />, known as Hartogs radii. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h04664013.png" />, when all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046640/h04664014.png" /> are constant, a Hartogs–Laurent series is called a [[Laurent series|Laurent series]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,   "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Behnke,   P. Thullen,   "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer  (1970)  (Elraged &amp; Revised Edition. Original: 1934)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Bochner,   W.T. Martin,   "Several complex variables" , Princeton Univ. Press  (1948)</TD></TR></table>