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

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

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

Then the radius of convergence $R$ of f at the point a is given by

{\frac {1}{R}}=\limsup _{n\to \infty }\left(|c_{n}|^{1/n}\right)

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 ${\textstyle \sum _{n}c_{n}z^{n}}$ 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 $\varepsilon >0$ , there exists only a finite number of $n$ such that ${\textstyle {\sqrt[{n}]{|c_{n}|}}\geq t+\varepsilon }$ . Now $|c_{n}|\leq (t+\varepsilon )^{n}$ for all but a finite number of $c_{n}$ , so the series ${\textstyle \sum _{n}c_{n}z^{n}}$ converges if $|z|<1/(t+\varepsilon )$ . This proves the first part.

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

Theorem for several complex variables

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

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

to the multidimensional power series

\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}}

Proof

From ^[6]

Set $z=a+t\rho$ $(z_{i}=a_{i}+t\rho _{i})$ , then

\sum _{\alpha \geq 0}c_{\alpha }(z-a)^{\alpha }=\sum _{\alpha \geq 0}c_{\alpha }\rho ^{\alpha }t^{|\alpha |}=\sum _{\mu \geq 0}\left(\sum _{|\alpha |=\mu }|c_{\alpha }|\rho ^{\alpha }\right)t^{\mu }

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

\limsup _{\mu \to \infty }{\sqrt[{\mu }]{\sum _{|\alpha |=\mu }|c_{\alpha }|\rho ^{\alpha }}}=1

Setting $|c_{m}|\rho ^{m}=\max _{|\alpha |=\mu }|c_{\alpha }|\rho ^{\alpha }$ gives us an estimate

|c_{m}|\rho ^{m}\leq \sum _{|\alpha |=\mu }|c_{\alpha }|\rho ^{\alpha }\leq (\mu +1)^{n}|c_{m}|\rho ^{m}

Because ${\sqrt[{\mu }]{(\mu +1)^{n}}}\to 1$ as $\mu \to \infty$

{\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\leq {\sqrt[{\mu }]{\sum _{|\alpha |=\mu }|c_{\alpha }|\rho ^{\alpha }}}\leq {\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\implies {\sqrt[{\mu }]{\sum _{|\alpha |=\mu }|c_{\alpha }|\rho ^{\alpha }}}={\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}\qquad (\mu \to \infty )

Therefore

\limsup _{|\alpha |\to \infty }{\sqrt[{|\alpha |}]{|c_{\alpha }|\rho ^{\alpha }}}=\limsup _{\mu \to \infty }{\sqrt[{\mu }]{|c_{m}|\rho ^{m}}}=1

Notes

↑ Cauchy, A. L. (1821), Analyse algébrique.
↑ 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.
↑ Hadamard, J., "Sur le rayon de convergence des séries ordonnées suivant les puissances d'une variable", C. R. Acad. Sci. Paris, 106: 259–262.
↑ 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.
↑ Lang, Serge (2002), Complex Analysis: Fourth Edition, Springer, pp. 55–56, ISBN 0-387-98592-1 Graduate Texts in Mathematics
↑ Shabat, B.V. (1992), Introduction to complex analysis Part II. Functions of several variables, American Mathematical Society, pp. 32–33, ISBN 978-0821819753

External links

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

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[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, pp. 32–33, ISBN 978-0821819753

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