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 R 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 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

Without loss of generality assume that a=0. We will show first that the power series converges for |z|<R, and then that it diverges for |z|>R. First suppose |z|<R. Let t=1/R not be 0 or \pm\infty. For any, there exists only a finite number of n such that. Now for all but a finite number of c_n, so the series converges if. This proves the first part. Conversely, for, for infinitely many c_n, so if , we see that the series cannot converge because its nth term does not tend to 0.

Theorem for several complex variables

Let \alpha be an n-dimensional vector of natural numbers with, then f(x) 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 t which converges for |t| < 1 and diverges for |t| > 1. Therefore, by the Cauchy-Hadamard theorem for one variable Setting gives us an estimate Because as Therefore