View source for Cauchy-Hadamard theorem

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>

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>

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

<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>

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.

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>

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:

<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>

<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>

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" />).

====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) <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" />:

<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>

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]]).

====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>