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.

## Theory for one complex variable

### Statement of the theorem

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

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

## Several complex variables

### Statement of the theorem

to the multidimensional power series

### 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