Cauchy-Hadamard theorem

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