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]

Theory for one complex variable[edit]

Statement of the theorem[edit]

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

where

Then the radius of convergence of ƒ 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 of the theorem[edit]

[5] 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 zero or ±infinity. 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.

Several complex variables[edit]

Statement of the theorem[edit]

Let be a multi-index (a n-tuple of integers) with , then converges with radius of convergence (which is also a multi-index) if and only if

to the multidimensional power series

Proof of the theorem[edit]

The proof can be found in the book Introduction to Complex Analysis Part II functions in several Variables by B. V. Shabat

Notes[edit]

  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

External links[edit]