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A 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>
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$$ \tag{* }
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\sum _ {k = - \infty } ^  \infty 
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f _ {k} ( \prime z)
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( z _ {n} - a _ {n} ) ^ {k} ,
 +
$$
  
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
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where $  \prime z = ( z _ {1} \dots z _ {n-} 1 ) $
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and where the $  f _ {k} ( \prime z) $
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are functions holomorphic in some domain $  \prime D \subset  \mathbf C  ^ {n-} 1 $
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which is independent of $  k $.  
 +
If $  f _ {k} = 0 $
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for all $  k < 0 $,  
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the series (*) is known as a Hartogs series. Any holomorphic function in a [[Hartogs domain|Hartogs domain]] $  D $
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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>
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$$
 +
\{ {( \prime z, z _ {n} ) } : {
 +
\prime z \in \prime D,\
 +
0 \leq  r ( \prime z) < | z _ {n} - a _ {n} | <
 +
R ( \prime z) \leq  + \infty } \}
 +
$$
  
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]].
+
can be expanded into a Hartogs–Laurent series which converges absolutely and uniformly inside $  D $.  
 +
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 $  r ( \prime z) $
 +
and $  R ( \prime z) $,  
 +
known as Hartogs radii. If $  n = 1 $,  
 +
when all $  f _ {k} $
 +
are constant, a Hartogs–Laurent series is called a [[Laurent series|Laurent series]].
  
 
====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:43, 5 June 2020


A series

$$ \tag{* } \sum _ {k = - \infty } ^ \infty f _ {k} ( \prime z) ( z _ {n} - a _ {n} ) ^ {k} , $$

where $ \prime z = ( z _ {1} \dots z _ {n-} 1 ) $ and where the $ f _ {k} ( \prime z) $ are functions holomorphic in some domain $ \prime D \subset \mathbf C ^ {n-} 1 $ which is independent of $ k $. If $ f _ {k} = 0 $ for all $ k < 0 $, the series (*) is known as a Hartogs series. Any holomorphic function in a Hartogs domain $ D $ of the type

$$ \{ {( \prime z, z _ {n} ) } : { \prime z \in \prime D,\ 0 \leq r ( \prime z) < | z _ {n} - a _ {n} | < R ( \prime z) \leq + \infty } \} $$

can be expanded into a Hartogs–Laurent series which converges absolutely and uniformly inside $ D $. 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 $ r ( \prime z) $ and $ R ( \prime z) $, known as Hartogs radii. If $ n = 1 $, when all $ f _ {k} $ are constant, a Hartogs–Laurent series is called a Laurent series.

References

[1] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)

Comments

References

[a1] H. Behnke, P. Thullen, "Theorie der Funktionen meherer komplexer Veränderlichen" , Springer (1970) (Elraged & Revised Edition. Original: 1934)
[a2] S. Bochner, W.T. Martin, "Several complex variables" , Princeton Univ. Press (1948)
How to Cite This Entry:
Hartogs-Laurent series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hartogs-Laurent_series&oldid=47189
This article was adapted from an original article by E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article