Cauchy-Hadamard theorem

Consider a complex power series

(1)

and let

If , then the series (1) is convergent only at the point ; if , then the series (1) is absolutely convergent in the disc where

(2)

and divergent outside the disc, where ; if , the series (1) is absolutely convergent for all . 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 and . 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 of radius (2). In the case of a real power series (1), formula (2) defines the "radius" of the interval of convergence: . 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

(3)

in complex variables , , one has the following generalization of the Cauchy–Hadamard formula:

(4)

which is valid for the associated radii of convergence of the series (3) (see Disc of convergence). Writing (4) in the form , one obtains an equation defining the boundary of a certain logarithmically convex Reinhardt domain with centre , which is the interior of the set of points at which the series (3) is absolutely convergent ().

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) and suppose that for some constant , some and for all :

Then the power series in (3) converges absolutely in the polydisc (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