Cauchy–Hadamard theorem
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[]
Consider the formal power series in one complex variable z of the form
Then the radius of convergence of f at the point a is given by
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.[5]
Theorem for several complex variables[]
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
The proof can be found in [6]
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, 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.
- ^ 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, ISBN 978-0821819753
External links[]
- Augustin-Louis Cauchy
- Mathematical series
- Theorems in complex analysis