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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 [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)

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).