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|, and then that it diverges for $|z|>R$.

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

Conversely, for $\epsilon > 0$, $|c_n|\geq (t-\epsilon)^n$ for infinitely many $c_n$, so if $|z|=1/(t-\epsilon) > R$, 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 $|\alpha|=\alpha_1+\ldots+\alpha_n$, 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

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

### 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. ^ . 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