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Cauchy–Hadamard theorem

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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, 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.

Theorem for one complex variable

Consider the formal power series in one complex variable z of the form f ( z ) = n = 0 c n ( z a ) n {\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-a)^{n}} where a , c n C . {\displaystyle a,c_{n}\in \mathbb {C} .}

Then the radius of convergence R {\displaystyle R} of f at the point a is given by 1 R = lim sup n ( | c n | 1 / n ) {\displaystyle {\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 is 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 {\displaystyle a=0} . We will show first that the power series n c n z n {\textstyle \sum _{n}c_{n}z^{n}} converges for | z | < R {\displaystyle |z|<R} , and then that it diverges for | z | > R {\displaystyle |z|>R} .

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

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

Theorem for several complex variables

Let α {\displaystyle \alpha } be an n-dimensional vector of natural numbers ( α = ( α 1 , , α n ) N n {\displaystyle \alpha =(\alpha _{1},\cdots ,\alpha _{n})\in \mathbb {N} ^{n}} ) with α = α 1 + + α n {\displaystyle \|\alpha \|=\alpha _{1}+\cdots +\alpha _{n}} , then f ( x ) {\displaystyle f(x)} converges with radius of convergence ρ = ( ρ 1 , , ρ n ) R n {\displaystyle \rho =(\rho _{1},\cdots ,\rho _{n})\in \mathbb {R} ^{n}} with ρ α = ρ 1 α 1 ρ n α n {\displaystyle \rho ^{\alpha }=\rho _{1}^{\alpha _{1}}\cdots \rho _{n}^{\alpha _{n}}} if and only if lim sup α | c α | ρ α α = 1 {\displaystyle \limsup _{\|\alpha \|\to \infty }{\sqrt{|c_{\alpha }|\rho ^{\alpha }}}=1} to the multidimensional power series α 0 c α ( z a ) α := α 1 0 , , α n 0 c α 1 , , α n ( z 1 a 1 ) α 1 ( z n a n ) α n . {\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}}.}

Proof

From

Set z = a + t ρ {\displaystyle z=a+t\rho } ( z i = a i + t ρ i ) . {\displaystyle (z_{i}=a_{i}+t\rho _{i}).} Then

α 0 c α ( z a ) α = α 0 c α ρ α t α = μ 0 ( α = μ | c α | ρ α ) t μ . {\displaystyle \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 {\displaystyle t} which converges for | t | < 1 {\displaystyle |t|<1} and diverges for | t | > 1 {\displaystyle |t|>1} . Therefore, by the Cauchy–Hadamard theorem for one variable

lim sup μ α = μ | c α | ρ α μ = 1. {\displaystyle \limsup _{\mu \to \infty }{\sqrt{\sum _{\|\alpha \|=\mu }|c_{\alpha }|\rho ^{\alpha }}}=1.}

Setting | c m | ρ m = max α = μ | c α | ρ α {\displaystyle |c_{m}|\rho ^{m}=\max _{\|\alpha \|=\mu }|c_{\alpha }|\rho ^{\alpha }} gives us an estimate

| c m | ρ m α = μ | c α | ρ α ( μ + 1 ) n | c m | ρ m . {\displaystyle |c_{m}|\rho ^{m}\leq \sum _{\|\alpha \|=\mu }|c_{\alpha }|\rho ^{\alpha }\leq (\mu +1)^{n}|c_{m}|\rho ^{m}.}

Because ( μ + 1 ) n μ 1 {\displaystyle {\sqrt{(\mu +1)^{n}}}\to 1} as μ {\displaystyle \mu \to \infty }

| c m | ρ m μ α = μ | c α | ρ α μ | c m | ρ m μ α = μ | c α | ρ α μ = | c m | ρ m μ ( μ ) . {\displaystyle {\sqrt{|c_{m}|\rho ^{m}}}\leq {\sqrt{\sum _{\|\alpha \|=\mu }|c_{\alpha }|\rho ^{\alpha }}}\leq {\sqrt{|c_{m}|\rho ^{m}}}\implies {\sqrt{\sum _{\|\alpha \|=\mu }|c_{\alpha }|\rho ^{\alpha }}}={\sqrt{|c_{m}|\rho ^{m}}}\qquad (\mu \to \infty ).}

Therefore

lim sup α | c α | ρ α α = lim sup μ | c m | ρ m μ = 1. {\displaystyle \limsup _{\|\alpha \|\to \infty }{\sqrt{|c_{\alpha }|\rho ^{\alpha }}}=\limsup _{\mu \to \infty }{\sqrt{|c_{m}|\rho ^{m}}}=1.}

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 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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