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]

Theorem for one complex variable

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

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

where ${\displaystyle a,c_{n}\in \mathbb {C} .}$

Then the radius of convergence ${\displaystyle R}$ of ƒ at the point a is given by

${\displaystyle {\frac {1}{R}}=\limsup _{n\to \infty }{\big (}|c_{n}|^{1/n}{\big )}}$

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

Without loss of generality assume that ${\displaystyle a=0}$. We will show first that the power series ${\displaystyle \sum c_{n}z^{n}}$ converges for ${\displaystyle |z|<R}$, and then that it diverges for ${\displaystyle |z|>R}$.

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

Conversely, for ${\displaystyle \varepsilon >0}$, ${\displaystyle |c_{n}|\geq (t-\varepsilon )^{n}}$ for infinitely many ${\displaystyle c_{n}}$, so if ${\displaystyle |z|=1/(t-\varepsilon )>R}$, we see that the series cannot converge because its nth term does not tend to 0.[5]

Theorem for several complex variables

Let ${\displaystyle \alpha }$ be a multi-index (a n-tuple of integers) with ${\displaystyle |\alpha |=\alpha _{1}+\cdots +\alpha _{n}}$, then ${\displaystyle f(x)}$ converges with radius of convergence ${\displaystyle \rho }$ (which is also a multi-index) if and only if

${\displaystyle \lim _{|\alpha |\to \infty }{\sqrt[{|\alpha |}]{|c_{\alpha }|\rho ^{\alpha }}}=1}$

to the multidimensional power series

${\displaystyle \sum _{\alpha \geq 0}c_{\alpha }(z-a)^{\alpha }:=\sum _{\alpha _{1}\geq 0,\ldots ,\alpha _{n}\geq 0}c_{\alpha _{1},\ldots ,\alpha _{n}}(z_{1}-a_{1})^{\alpha _{1}}\cdots (z_{n}-a_{n})^{\alpha _{n}}}$

The proof can be found in [6]

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 0-387-98592-1 Graduate Texts in Mathematics
6. Shabat, B.V. (1992), Introduction to complex analysis Part II. Functions of several variables, American Mathematical Society, ISBN 978-0821819753

External links

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