View source for Hartogs domain

''semi-circular domain, with symmetry plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046630/h0466301.png" />''

A domain in the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046630/h0466302.png" /> complex variables which, for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046630/h0466303.png" />, contains the circle

<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/h046630/h0466304.png" /></td> </tr></table>

Named after F. Hartogs. A Hartogs domain is called complete if for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046630/h0466305.png" /> it contains the disc

<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/h046630/h0466306.png" /></td> </tr></table>

A Hartogs domain with symmetry plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046630/h0466307.png" /> can conveniently be represented by a Hartogs diagram, viz., by the image of the Hartogs domain under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046630/h0466308.png" />.

====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><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Bochner,   W.T. Martin,   "Several complex variables" , Princeton Univ. Press  (1948)</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></table>