A theorem that determines the radius of convergence of a power series.
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, but remained relatively unknown until Hadamard rediscovered it. Hadamard's first publication of this result was in 1888; he also included it as part of his 1892 Ph.D. thesis.
Theorem for one complex variable
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 is 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
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.
Theorem for several complex variables
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
From
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
- Cauchy, A. L. (1821), Analyse algébrique.
- 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.
- Hadamard, J., "Sur le rayon de convergence des séries ordonnées suivant les puissances d'une variable", C. R. Acad. Sci. Paris, 106: 259–262.
- Hadamard, J. (1892), "Essai sur l'étude des fonctions données par leur développement de Taylor", Journal de Mathématiques Pures et Appliquées, 4 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.
- Lang, Serge (2002), Complex Analysis: Fourth Edition, Springer, pp. 55–56, ISBN 0-387-98592-1 Graduate Texts in Mathematics
- 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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