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Difference between revisions of "Hartogs domain"

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''semi-circular domain, with symmetry plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046630/h0466301.png" />''
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''semi-circular domain, with symmetry plane $\{z_n=a_n\}$''
  
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
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A domain in the space of $n$ complex variables which, for each point $z=(z_1,\dots,z_{n-1},z_n)\equiv('z,z_n)$, 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>
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$$\left\{('z,a_n+e^{i\theta}(z_n-a_n)):0\leq\theta<2\pi\right\}.$$
  
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
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Named after F. Hartogs. A Hartogs domain is called complete if for each point $('z,z_n)$ 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>
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$$\{('z,a_n+\lambda(z_n-a_n)):|\lambda|\leq1\}.$$
  
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" />.
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A Hartogs domain with symmetry plane $\{z_n=0\}$ can conveniently be represented by a Hartogs diagram, viz., by the image of the Hartogs domain under the mapping $('z,z_n)\to('z,|z_n|)$.
  
 
====References====
 
====References====

Latest revision as of 20:11, 22 November 2018

semi-circular domain, with symmetry plane $\{z_n=a_n\}$

A domain in the space of $n$ complex variables which, for each point $z=(z_1,\dots,z_{n-1},z_n)\equiv('z,z_n)$, contains the circle

$$\left\{('z,a_n+e^{i\theta}(z_n-a_n)):0\leq\theta<2\pi\right\}.$$

Named after F. Hartogs. A Hartogs domain is called complete if for each point $('z,z_n)$ it contains the disc

$$\{('z,a_n+\lambda(z_n-a_n)):|\lambda|\leq1\}.$$

A Hartogs domain with symmetry plane $\{z_n=0\}$ can conveniently be represented by a Hartogs diagram, viz., by the image of the Hartogs domain under the mapping $('z,z_n)\to('z,|z_n|)$.

References

[1] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[2] S. Bochner, W.T. Martin, "Several complex variables" , Princeton Univ. Press (1948)


Comments

References

[a1] H. Behnke, P. Thullen, "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer (1970) (Elraged & Revised Edition. Original: 1934)
How to Cite This Entry:
Hartogs domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hartogs_domain&oldid=15532
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article