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]

Theory for one complex variable

Statement of the theorem

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

f(z)=\sum _{{n=0}}^{{\infty }}c_{{n}}(z-a)^{{n}}

where $a,c_{n}\in {\mathbb {C}}.$

Then the radius of convergence of ƒ at the point a is given by

{\frac {1}{R}}=\limsup _{{n\to \infty }}{\big (}|c_{{n}}|^{{1/n}}{\big )}

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

^[5] Without loss of generality assume that $a=0$ . We will show first that the power series $\sum c_{n}z^{n}$ converges for $|z|<R$ , and then that it diverges for $|z|>R$ .

First suppose $|z|<R$ . Let $t=1/R$ not be zero or ±infinity. For any $\varepsilon >0$ , there exists only a finite number of $n$ such that ${\sqrt[{n}]{|c_{n}|}}\geq t+\varepsilon$ . Now $|c_{n}|\leq (t+\varepsilon )^{n}$ for all but a finite number of $n$ , so the series $\sum c_{n}z^{n}$ converges if $|z|<1/(t+\varepsilon )$ . This proves the first part.

Conversely, for $\varepsilon >0$ , $|c_{n}|\geq (t-\varepsilon )^{n}$ for infinitely many $c_{n}$ , so if $|z|=1/(t-\varepsilon )>R$ , we see that the series cannot converge because its nth term does not tend to 0.

Several complex variables

Statement of the theorem

Let $\alpha$ be a multi-index (a n-tuple of integers) with $|\alpha |=\alpha _{1}+\cdots +\alpha _{n}$ , then $f(x)$ converges with radius of convergence $\rho$ (which is also a multi-index) if and only if

\lim _{{|\alpha |\to \infty }}{\sqrt[ {|\alpha |}]{|c_{\alpha }|\rho ^{\alpha }}}=1

to the multidimensional power series

\sum _{\alpha \geq 0}c_{\alpha }(z-a)^{\alpha }:=\sum _{\alpha _{1}\geq 0,\ldots ,\alpha _{n}\geq 0}c_{\alpha _{1},\ldots ,\alpha _{n}}(z_{1}-a_{1})^{\alpha _{1}}\cdots (z_{n}-a_{n})^{\alpha _{n}}

Proof of the theorem

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

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^e 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

External links

Weisstein, Eric Wolfgang. "Cauchy-Hadamard theorem". MathWorld.

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