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Cauchy–Hadamard theorem

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In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy,[1] but remained relatively unknown until Hadamard rediscovered it.[2] Hadamard's first publication of this result was in 1888;[3] he also included it as part of his 1892 Ph.D. thesis.[4]

Theorem for one complex variable

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Consider the formal power series in one complex variable z of the form where

Then the radius of convergence of f at the point a is given by where lim sup denotes the limit superior, the limit as n approaches infinity of the supremum of the sequence values after the nth position. If the sequence values are unbounded so that the lim sup is ∞, then the power series does not converge near a, while if the lim sup is 0 then the radius of convergence is ∞, meaning that the series converges on the entire plane.

Proof

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Without loss of generality assume that . We will show first that the power series converges for , and then that it diverges for .

First suppose . Let not be or For any , there exists only a finite number of such that . Now for all but a finite number of , so the series converges if . This proves the first part.

Conversely, for , for infinitely many , so if , we see that the series cannot converge because its nth term does not tend to 0.[5]

Theorem for several complex variables

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Let be an n-dimensional vector of natural numbers () with , then converges with radius of convergence with if and only if to the multidimensional power series

Proof

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From [6]

Set , then

This is a power series in one variable which converges for and diverges for . Therefore, by the Cauchy-Hadamard theorem for one variable

Setting gives us an estimate

Because as

Therefore

Notes

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  1. ^ Cauchy, A. L. (1821), Analyse algébrique.
  2. ^ Bottazzini, Umberto (1986), The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer-Verlag, pp. 116–117, ISBN 978-0-387-96302-0. Translated from the Italian by Warren Van Egmond.
  3. ^ Hadamard, J., "Sur le rayon de convergence des séries ordonnées suivant les puissances d'une variable", C. R. Acad. Sci. Paris, 106: 259–262.
  4. ^ Hadamard, J. (1892), "Essai sur l'étude des fonctions données par leur développement de Taylor", Journal de Mathématiques Pures et Appliquées, 4e Série, VIII. Also in Thèses présentées à la faculté des sciences de Paris pour obtenir le grade de docteur ès sciences mathématiques, Paris: Gauthier-Villars et fils, 1892.
  5. ^ Lang, Serge (2002), Complex Analysis: Fourth Edition, Springer, pp. 55–56, ISBN 0-387-98592-1 Graduate Texts in Mathematics
  6. ^ Shabat, B.V. (1992), Introduction to complex analysis Part II. Functions of several variables, American Mathematical Society, pp. 32–33, ISBN 978-0821819753
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