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Consider a complex power series
 
Consider a complex 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/c/c020/c020870/c0208701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
f (z)  = \
 +
\sum _ {k = 0 } ^  \infty 
 +
c _ {k} (z - a) ^ {k}
 +
$$
  
 
and let
 
and let
  
<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/c/c020/c020870/c0208702.png" /></td> </tr></table>
+
$$
 +
\Lambda  = \
 +
\lim\limits _ {k \rightarrow \infty } \
 +
\sup \
 +
| c _ {k} |  ^ {1/k} .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c0208703.png" />, then the series (1) is convergent only at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c0208704.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c0208705.png" />, then the series (1) is absolutely convergent in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c0208706.png" /> where
+
If $  \Lambda = \infty $,  
 +
then the series (1) is convergent only at the point $  z = a $;  
 +
if $  0 < \Lambda < \infty $,  
 +
then the series (1) is absolutely convergent in the disc $  | z - a | < R $
 +
where
  
<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/c/c020/c020870/c0208707.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
= {
 +
\frac{1} \Lambda
 +
} ,
 +
$$
  
and divergent outside the disc, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c0208708.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c0208709.png" />, the series (1) is absolutely convergent for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087010.png" />. The content of the Cauchy–Hadamard theorem is thus expressed by the Cauchy–Hadamard formula (2), which should be understood in this context in a broad sense, including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087012.png" />. In other words, the Cauchy–Hadamard theorem states that the interior of the set of points at which the series (1) is (absolutely) convergent is the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087013.png" /> of radius (2). In the case of a real power series (1), formula (2) defines the  "radius"  of the interval of convergence: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087014.png" />. Essentially, the Cauchy–Hadamard theorem was stated by A.L. Cauchy in his lectures [[#References|[1]]] in 1821; it was J. Hadamard [[#References|[2]]] who made the formulation and the proof fully explicit.
+
and divergent outside the disc, where $  | z - a | > R $;  
 +
if $  \Lambda = 0 $,  
 +
the series (1) is absolutely convergent for all $  z \in \mathbf C $.  
 +
The content of the Cauchy–Hadamard theorem is thus expressed by the Cauchy–Hadamard formula (2), which should be understood in this context in a broad sense, including $  1/ \infty = 0 $
 +
and  $  1/0 = \infty $.  
 +
In other words, the Cauchy–Hadamard theorem states that the interior of the set of points at which the series (1) is (absolutely) convergent is the disc $  | z - a | < R $
 +
of radius (2). In the case of a real power series (1), formula (2) defines the  "radius"  of the interval of convergence: $  a - R < x < a + R $.  
 +
Essentially, the Cauchy–Hadamard theorem was stated by A.L. Cauchy in his lectures [[#References|[1]]] in 1821; it was J. Hadamard [[#References|[2]]] who made the formulation and the proof fully explicit.
  
 
For power series
 
For 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/c/c020/c020870/c02087015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
f (z)  = \
 +
\sum _ {k _ {1} \dots k _ {n} = 0 } ^  \infty 
 +
c _ {k _ {1}  \dots k _ {n} }
 +
(z _ {1} - a _ {1} ) ^ {k _ {1} } \dots
 +
(z _ {n} - a _ {n} ) ^ {k _ {n} }
 +
$$
  
in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087016.png" /> complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087018.png" />, one has the following generalization of the Cauchy–Hadamard formula:
+
in $  n $
 +
complex variables $  z = (z _ {1} \dots z _ {n} ) $,  
 +
$  n \geq  1 $,  
 +
one has the following generalization of the Cauchy–Hadamard formula:
  
<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/c/c020/c020870/c02087019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\lim\limits _ {| k | \rightarrow \infty } \
 +
| c _ {k _ {1}  \dots k _ {n} } | ^ {1 / | k | }
 +
r _ {1} ^ {k _ {1} } \dots r _ {n} ^ {k _ {n} }
 +
= 1,
 +
$$
  
<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/c/c020/c020870/c02087020.png" /></td> </tr></table>
+
$$
 +
| k |  = k _ {1} + \dots + k _ {n} ,
 +
$$
  
which is valid for the associated radii of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087021.png" /> of the series (3) (see [[Disc of convergence|Disc of convergence]]). Writing (4) in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087022.png" />, one obtains an equation defining the boundary of a certain logarithmically convex [[Reinhardt domain|Reinhardt domain]] with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087023.png" />, which is the interior of the set of points at which the series (3) is absolutely convergent (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087024.png" />).
+
which is valid for the associated radii of convergence $  r _ {1} \dots r _ {n} $
 +
of the series (3) (see [[Disc of convergence|Disc of convergence]]). Writing (4) in the form $  \Phi (r _ {1} \dots r _ {n} ) = 0 $,  
 +
one obtains an equation defining the boundary of a certain logarithmically convex [[Reinhardt domain|Reinhardt domain]] with centre $  a $,  
 +
which is the interior of the set of points at which the series (3) is absolutely convergent ( $  n > 1 $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.L. Cauchy,  "Analyse algébrique" , Gauthier-Villars , Leipzig  (1894)  (German translation: Springer, 1885)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Hadamard,  "Essai sur l'etude des fonctions données par leur développement de Taylor"  ''J. Math. Pures Appl.'' , '''8''' :  4  (1892)  pp. 101–186  (Thesis)</TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.L. Cauchy,  "Analyse algébrique" , Gauthier-Villars , Leipzig  (1894)  (German translation: Springer, 1885)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Hadamard,  "Essai sur l'etude des fonctions données par leur développement de Taylor"  ''J. Math. Pures Appl.'' , '''8''' :  4  (1892)  pp. 101–186  (Thesis)</TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The Cauchy–Hadamard theorem is related to Abel's lemma: Let in (3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087025.png" /> and suppose that for some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087026.png" />, some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087027.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087028.png" />:
+
The Cauchy–Hadamard theorem is related to Abel's lemma: Let in (3) $  a _ {1} = \dots = a _ {n} = 0 $
 +
and suppose that for some constant $  a $,  
 +
some $  \omega \in \mathbf C  ^ {n} $
 +
and for all $  k _ {1} \dots k _ {n} \geq  0 $:
  
<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/c/c020/c020870/c02087029.png" /></td> </tr></table>
+
$$
 +
| c _ {k _ {1}  \dots k _ {n} }
 +
w _ {1} ^ {k _ {1} } \dots w _ {n} ^ {k _ {n} } |
 +
\leq  A .
 +
$$
  
Then the power series in (3) converges absolutely in the [[Polydisc|polydisc]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020870/c02087030.png" /> (see [[#References|[a1]]], Sect. 2.4). This fact makes power series an effective tool in the [[Analytic continuation|analytic continuation]] of analytic functions of several variables (see also [[Hartogs theorem|Hartogs theorem]]).
+
Then the power series in (3) converges absolutely in the [[Polydisc|polydisc]] $  \{ {z \in \mathbf C  ^ {n} } : {| z _ {j} | < | w _ {j} | ,  j = 1 \dots n } \} $(
 +
see [[#References|[a1]]], Sect. 2.4). This fact makes power series an effective tool in the [[Analytic continuation|analytic continuation]] of analytic functions of several variables (see also [[Hartogs theorem|Hartogs theorem]]).
  
 
====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></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></table>

Latest revision as of 15:35, 4 June 2020


Consider a complex power series

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

and let

$$ \Lambda = \ \lim\limits _ {k \rightarrow \infty } \ \sup \ | c _ {k} | ^ {1/k} . $$

If $ \Lambda = \infty $, then the series (1) is convergent only at the point $ z = a $; if $ 0 < \Lambda < \infty $, then the series (1) is absolutely convergent in the disc $ | z - a | < R $ where

$$ \tag{2 } R = { \frac{1} \Lambda } , $$

and divergent outside the disc, where $ | z - a | > R $; if $ \Lambda = 0 $, the series (1) is absolutely convergent for all $ z \in \mathbf C $. The content of the Cauchy–Hadamard theorem is thus expressed by the Cauchy–Hadamard formula (2), which should be understood in this context in a broad sense, including $ 1/ \infty = 0 $ and $ 1/0 = \infty $. In other words, the Cauchy–Hadamard theorem states that the interior of the set of points at which the series (1) is (absolutely) convergent is the disc $ | z - a | < R $ of radius (2). In the case of a real power series (1), formula (2) defines the "radius" of the interval of convergence: $ a - R < x < a + R $. Essentially, the Cauchy–Hadamard theorem was stated by A.L. Cauchy in his lectures [1] in 1821; it was J. Hadamard [2] who made the formulation and the proof fully explicit.

For power series

$$ \tag{3 } f (z) = \ \sum _ {k _ {1} \dots k _ {n} = 0 } ^ \infty c _ {k _ {1} \dots k _ {n} } (z _ {1} - a _ {1} ) ^ {k _ {1} } \dots (z _ {n} - a _ {n} ) ^ {k _ {n} } $$

in $ n $ complex variables $ z = (z _ {1} \dots z _ {n} ) $, $ n \geq 1 $, one has the following generalization of the Cauchy–Hadamard formula:

$$ \tag{4 } \lim\limits _ {| k | \rightarrow \infty } \ | c _ {k _ {1} \dots k _ {n} } | ^ {1 / | k | } r _ {1} ^ {k _ {1} } \dots r _ {n} ^ {k _ {n} } = 1, $$

$$ | k | = k _ {1} + \dots + k _ {n} , $$

which is valid for the associated radii of convergence $ r _ {1} \dots r _ {n} $ of the series (3) (see Disc of convergence). Writing (4) in the form $ \Phi (r _ {1} \dots r _ {n} ) = 0 $, one obtains an equation defining the boundary of a certain logarithmically convex Reinhardt domain with centre $ a $, which is the interior of the set of points at which the series (3) is absolutely convergent ( $ n > 1 $).

References

[1] A.L. Cauchy, "Analyse algébrique" , Gauthier-Villars , Leipzig (1894) (German translation: Springer, 1885)
[2] J. Hadamard, "Essai sur l'etude des fonctions données par leur développement de Taylor" J. Math. Pures Appl. , 8 : 4 (1892) pp. 101–186 (Thesis)
[3] A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[4] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)

Comments

The Cauchy–Hadamard theorem is related to Abel's lemma: Let in (3) $ a _ {1} = \dots = a _ {n} = 0 $ and suppose that for some constant $ a $, some $ \omega \in \mathbf C ^ {n} $ and for all $ k _ {1} \dots k _ {n} \geq 0 $:

$$ | c _ {k _ {1} \dots k _ {n} } w _ {1} ^ {k _ {1} } \dots w _ {n} ^ {k _ {n} } | \leq A . $$

Then the power series in (3) converges absolutely in the polydisc $ \{ {z \in \mathbf C ^ {n} } : {| z _ {j} | < | w _ {j} | , j = 1 \dots n } \} $( see [a1], Sect. 2.4). This fact makes power series an effective tool in the analytic continuation of analytic functions of several variables (see also Hartogs theorem).

References

[a1] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4
How to Cite This Entry:
Cauchy-Hadamard theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy-Hadamard_theorem&oldid=22259
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article