# 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

Then the radius of convergence of *ƒ* 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 *n*th 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 . We will show first that the power series converges for , and then that it diverges for .

First suppose . Let not be zero or ±infinity. For any , there exists only a finite number of such that

. Now for all but a finite number of , so the series converges if . This proves the first part.

Conversely, for , for infinitely many , so if , we see that the series cannot converge because its *n*th term does not tend to 0.

## Several complex variables

### Statement of the theorem

Let be a multi-index (a *n*-tuple of integers) with , then converges with radius of convergence (which is also a multi-index) if and only if

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

## Notes

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*Thèses présentées à la faculté des sciences de Paris pour obtenir le grade de docteur ès sciences mathématiques*, Paris: Gauthier-Villars et fils, 1892. - ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}Graduate Texts in Mathematics