Namespaces
Variants
Actions

Difference between revisions of "Disc of convergence"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
d0329701.png
 +
$#A+1 = 36 n = 0
 +
$#C+1 = 36 : ~/encyclopedia/old_files/data/D032/D.0302970 Disc of convergence
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''of a power series
 
''of a power 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/d/d032/d032970/d0329701.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
f ( z)  = \
 +
\sum _ {k = 0 } ^  \infty 
 +
c _ {k} ( z - a)  ^ {k}
 +
$$
  
 
''
 
''
  
The disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d0329702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d0329703.png" />, in which the series
+
The disc $  \Delta = \{ {z } : {| z - a | < R } \} $,  
 +
$  z \in \mathbf C $,  
 +
in which the series
  
is absolutely convergent, while outside the disc (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d0329704.png" />) it is divergent. In other words, the disc of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d0329705.png" /> is the interior of the set of points of convergence of the series . Its radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d0329706.png" /> is called the radius of convergence of the series. The disc of convergence may shrink to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d0329707.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d0329708.png" />, and it may be the entire open plane, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d0329709.png" />. The radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297010.png" /> is equal to the distance of the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297011.png" /> to the set of singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297012.png" /> (for the determination of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297013.png" /> in terms of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297014.png" /> of the series see [[Cauchy–Hadamard theorem|Cauchy–Hadamard theorem]]). Any disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297016.png" />, in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297017.png" />-plane is the disc of convergence of some power series.
+
is absolutely convergent, while outside the disc (for $  | z - a | > R $)  
 +
it is divergent. In other words, the disc of convergence $  \Delta $
 +
is the interior of the set of points of convergence of the series . Its radius $  R $
 +
is called the radius of convergence of the series. The disc of convergence may shrink to the point $  a $
 +
when $  R = 0 $,  
 +
and it may be the entire open plane, when $  R = \infty $.  
 +
The radius of convergence $  R $
 +
is equal to the distance of the centre $  a $
 +
to the set of singular points of $  f ( z) $(
 +
for the determination of $  R $
 +
in terms of the coefficients $  c _ {k} $
 +
of the series see [[Cauchy–Hadamard theorem|Cauchy–Hadamard theorem]]). Any disc $  \Delta = \{ {z } : {| z | < R } \} $,  
 +
0 \leq  R \leq  \infty $,  
 +
in the $  z $-
 +
plane is the disc of convergence of some power series.
  
 
For a power series
 
For a power 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/d/d032/d032970/d03297018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
f ( z)  = \
 +
f ( z _ {1} \dots z _ {n} ) =
 +
$$
  
<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/d/d032/d032970/d03297019.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {k _ {1} \dots k _ {n} = 0 } ^  \infty  c _ {k _ {1} \dots k _ {n}  } ( z _ {1} - a _ {1} ) ^ {k _ {1} } \dots ( z _ {n} - a _ {n} ) ^ {k _ {n} }
 +
$$
  
in several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297021.png" />, a polydisc of convergence of the series (2) is defined to be any polydisc
+
in several complex variables $  z _ {1} \dots z _ {n} $,  
 +
$  n > 1 $,  
 +
a polydisc of convergence of the series (2) is defined to be any polydisc
  
<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/d/d032/d032970/d03297022.png" /></td> </tr></table>
+
$$
 +
\Delta _ {n}  = \
 +
\{ {z = ( z _ {1} \dots z _ {n} ) } : {
 +
| z _  \nu  - a _  \nu  | < R _  \nu  ,\
 +
\nu = 1 \dots n } \}
 +
$$
  
 
at all points of which the series (2) is absolutely convergent, while in any polydisc
 
at all points of which the series (2) is absolutely convergent, while in any polydisc
  
<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/d/d032/d032970/d03297023.png" /></td> </tr></table>
+
$$
 +
\{ {z = ( z _ {1} \dots z _ {n} ) } : {
 +
| z _  \nu  - a _  \nu  | < R _  \nu  ^ { \prime } ,\
 +
\nu = 1 \dots n } \}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297024.png" /> and at least one of the latter inequalities is strict, there is at least one point at which the series is divergent. The radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297027.png" />, of the polydisc of convergence are called the associated radii of convergence of the series (2). They are in a well-defined relationship with the coefficients of the series, so that any polydisc with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297028.png" /> and with radii satisfying this relationship is the polydisc of convergence of a series (2) (cf. [[Cauchy–Hadamard theorem|Cauchy–Hadamard theorem]]). Any polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297031.png" />, in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297032.png" /> is the polydisc of convergence for some power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297033.png" /> complex variables. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297034.png" /> the interior of the set of points of absolute convergence of a series (2) is more complicated — it is a logarithmically convex complete Reinhardt domain with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297035.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032970/d03297036.png" /> (cf. [[Reinhardt domain|Reinhardt domain]]).
+
where $  R _  \nu  ^ { \prime } \geq  R _  \nu  $
 +
and at least one of the latter inequalities is strict, there is at least one point at which the series is divergent. The radii $  R _  \nu  $,  
 +
$  \nu = 1 \dots n $,  
 +
0 \leq  R _  \nu  \leq  \infty $,  
 +
of the polydisc of convergence are called the associated radii of convergence of the series (2). They are in a well-defined relationship with the coefficients of the series, so that any polydisc with centre $  a $
 +
and with radii satisfying this relationship is the polydisc of convergence of a series (2) (cf. [[Cauchy–Hadamard theorem|Cauchy–Hadamard theorem]]). Any polydisc $  \Delta _ {n} $,  
 +
0 \leq  R _  \nu  \leq  \infty $,  
 +
$  \nu = 1 \dots n $,  
 +
in the complex space $  \mathbf C  ^ {n} $
 +
is the polydisc of convergence for some power series in $  n $
 +
complex variables. When $  n > 1 $
 +
the interior of the set of points of absolute convergence of a series (2) is more complicated — it is a logarithmically convex complete Reinhardt domain with centre $  a $
 +
in $  \mathbf C  ^ {n} $(
 +
cf. [[Reinhardt domain|Reinhardt domain]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''1''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Hörmander,  "An introduction to complex analysis in several variables" , North-Holland  (1973)  pp. Chapt. 2.4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Ahlfors,  "Complex analysis" , McGraw-Hill  (1978)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1987)  pp. 24</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Hörmander,  "An introduction to complex analysis in several variables" , North-Holland  (1973)  pp. Chapt. 2.4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Ahlfors,  "Complex analysis" , McGraw-Hill  (1978)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1987)  pp. 24</TD></TR></table>

Latest revision as of 19:35, 5 June 2020


of a power series

$$ \tag{1 } f ( z) = \ \sum _ {k = 0 } ^ \infty c _ {k} ( z - a) ^ {k} $$

The disc $ \Delta = \{ {z } : {| z - a | < R } \} $, $ z \in \mathbf C $, in which the series

is absolutely convergent, while outside the disc (for $ | z - a | > R $) it is divergent. In other words, the disc of convergence $ \Delta $ is the interior of the set of points of convergence of the series . Its radius $ R $ is called the radius of convergence of the series. The disc of convergence may shrink to the point $ a $ when $ R = 0 $, and it may be the entire open plane, when $ R = \infty $. The radius of convergence $ R $ is equal to the distance of the centre $ a $ to the set of singular points of $ f ( z) $( for the determination of $ R $ in terms of the coefficients $ c _ {k} $ of the series see Cauchy–Hadamard theorem). Any disc $ \Delta = \{ {z } : {| z | < R } \} $, $ 0 \leq R \leq \infty $, in the $ z $- plane is the disc of convergence of some power series.

For a power series

$$ \tag{2 } f ( z) = \ f ( z _ {1} \dots z _ {n} ) = $$

$$ = \ \sum _ {k _ {1} \dots k _ {n} = 0 } ^ \infty c _ {k _ {1} \dots k _ {n} } ( z _ {1} - a _ {1} ) ^ {k _ {1} } \dots ( z _ {n} - a _ {n} ) ^ {k _ {n} } $$

in several complex variables $ z _ {1} \dots z _ {n} $, $ n > 1 $, a polydisc of convergence of the series (2) is defined to be any polydisc

$$ \Delta _ {n} = \ \{ {z = ( z _ {1} \dots z _ {n} ) } : { | z _ \nu - a _ \nu | < R _ \nu ,\ \nu = 1 \dots n } \} $$

at all points of which the series (2) is absolutely convergent, while in any polydisc

$$ \{ {z = ( z _ {1} \dots z _ {n} ) } : { | z _ \nu - a _ \nu | < R _ \nu ^ { \prime } ,\ \nu = 1 \dots n } \} , $$

where $ R _ \nu ^ { \prime } \geq R _ \nu $ and at least one of the latter inequalities is strict, there is at least one point at which the series is divergent. The radii $ R _ \nu $, $ \nu = 1 \dots n $, $ 0 \leq R _ \nu \leq \infty $, of the polydisc of convergence are called the associated radii of convergence of the series (2). They are in a well-defined relationship with the coefficients of the series, so that any polydisc with centre $ a $ and with radii satisfying this relationship is the polydisc of convergence of a series (2) (cf. Cauchy–Hadamard theorem). Any polydisc $ \Delta _ {n} $, $ 0 \leq R _ \nu \leq \infty $, $ \nu = 1 \dots n $, in the complex space $ \mathbf C ^ {n} $ is the polydisc of convergence for some power series in $ n $ complex variables. When $ n > 1 $ the interior of the set of points of absolute convergence of a series (2) is more complicated — it is a logarithmically convex complete Reinhardt domain with centre $ a $ in $ \mathbf C ^ {n} $( cf. Reinhardt domain).

References

[1] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[2] B.V. Shabat, "Introduction of complex analysis" , 1 , Moscow (1976) (In Russian)

Comments

References

[a1] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4
[a2] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1978)
[a3] W. Rudin, "Real and complex analysis" , McGraw-Hill (1987) pp. 24
How to Cite This Entry:
Disc of convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Disc_of_convergence&oldid=17451
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article