Cauchy–Hadamard theorem

{{Short description|A theorem that determines the radius of convergence of a power series.}}

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,{{citation|first=A. L.|last=Cauchy|author-link=Augustin Louis Cauchy|title=Analyse algébrique|year=1821}}. but remained relatively unknown until Hadamard rediscovered it.{{citation|title=The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass|first=Umberto|last=Bottazzini| publisher=Springer-Verlag|year=1986|isbn=978-0-387-96302-0|pages=[https://archive.org/details/highercalculushi0000bott/page/116 116–117]|url=https://archive.org/details/highercalculushi0000bott/page/116}}. Translated from the Italian by Warren Van Egmond. Hadamard's first publication of this result was in 1888;{{citation|first=J.|last=Hadamard|author-link=Jacques Hadamard|title=Sur le rayon de convergence des séries ordonnées suivant les puissances d'une variable|journal=C. R. Acad. Sci. Paris|volume=106|pages=259–262}}. he also included it as part of his 1892 Ph.D. thesis.{{citation| first=J.|last=Hadamard|author-link=Jacques Hadamard|title=Essai sur l'étude des fonctions données par leur développement de Taylor| url=https://archive.org/details/essaisurltuded00hadauoft| journal=Journal de Mathématiques Pures et Appliquées|series= 4^e Série|volume=VIII|year=1892}}. 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.

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 $a, c_n \in \Complex.$

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 {{math|lim sup}} denotes the limit superior, the limit as {{mvar|n}} approaches infinity of the supremum of the sequence values after the nth position. If the sequence values is unbounded so that the {{math|lim sup}} is ∞, then the power series does not converge near {{math|a}}, while if the {{math|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 $\sum_n c_n z^n$ converges for $|z|, and then that it diverges for |z|>R .$

First suppose $|z| . Let t=1/R not be 0 or \pm\infty.$

For any $\varepsilon > 0$ , there exists only a finite number of $n$ such that $\sqrt[n]$

c_n

\geq t+\varepsilon.

Now $|c_n| \leq (t+\varepsilon)^n$ for all but a finite number of $c_n$ , so the series $\sum_n c_n z^n$ 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.{{citation|title=Complex Analysis: Fourth Edition|first=Serge|last=Lang| publisher=Springer|year=2002| isbn=0-387-98592-1|pages=55–56}} Graduate Texts in Mathematics

Theorem for several complex variables

Let $\alpha$ be an n-dimensional vector of natural numbers ( $\alpha = (\alpha_1, \cdots, \alpha_n) \in \N^n$ ) with $\|\alpha\| := \alpha_1 + \cdots + \alpha_n$ , then $f(z)$ converges with radius of convergence $\rho = (\rho_1, \cdots, \rho_n) \in \R^n$ , $\rho^\alpha = \rho_1^{\alpha_1} \cdots \rho_n^{\alpha_n}$ if and only if

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

of the multidimensional power series

$f(z) = \sum_{\alpha\geq0}c_\alpha(z-a)^\alpha := \sum_{\alpha_1\geq0,\ldots,\alpha_n\geq0}c_{\alpha_1,\ldots,\alpha_n}(z_1-a_1)^{\alpha_1}\cdots(z_n-a_n)^{\alpha_n}.$

=Proof=

From {{citation| title=Introduction to complex analysis Part II. Functions of several variables| first=B. V.| last=Shabat| publisher=American Mathematical Society| year=1992| isbn=978-0821819753| pages=32–33}}

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

External links

{{MathWorld|urlname=Cauchy-HadamardTheorem|title=Cauchy-Hadamard theorem}}

Category:Augustin-Louis Cauchy

Category:Series (mathematics)

Category:Theorems in complex analysis