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 $c_n$, 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

$\sum_{\alpha\geq0}c_\alpha(z-a)^\alpha := \sum_{\alpha_1\geq0,\ldots,\alpha_n\geq0}c_{\alpha_1,\ldots,\alpha_n}(z_1-a_1)^{\alpha_1}\cdots(z_n-a_n)^{\alpha_n}$

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

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, 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.
5. Lang, Serge (2002), Complex Analysis: Fourth Edition, Springer, pp. 55–56, ISBN 0-387-98592-1Graduate Texts in Mathematics

External links

• Weisstein, Eric W., "Cauchy-Hadamard theorem", MathWorld.