View source for Cauchy-Hadamard theorem

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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 [[#References|[1]]] in 1821; it was J. Hadamard [[#References|[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|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====
<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====
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|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====
<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>