Cauchy–Hadamard theorem

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] Conteúdo 1 Theorem for one complex variable 1.1 Prova 2 Theorem for several complex variables 3 Notas 4 External links Theorem for one complex variable Consider the formal power series in one complex variable z of the form {estilo de exibição f(z)=soma _{n=0}^{infty }c_{n}(z-a)^{n}} Onde {estilo de exibição a,c_{n}em matemática {C} .} Then the radius of convergence {estilo de exibição R} of f at the point a is given by {estilo de exibição {fratura {1}{R}}=limsup_{até o infinito }deixei(|c_{n}|^{1/n}certo)} 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 {textstyle sum _{n}c_{n}z^{n}} converges for {estilo de exibição |z|R} .

First suppose {estilo de exibição |z|0} , there exists only a finite number of {estilo de exibição m} de tal modo que {estilo de texto {quadrado[{n}]{|c_{n}|}}geq t+varepsilon } . Agora {estilo de exibição |c_{n}|leq (t+varepsilon )^{n}} for all but a finite number of {estilo de exibição c_{n}} , so the series {textstyle sum _{n}c_{n}z^{n}} converges if {estilo de exibição |z|<1/(t+varepsilon )} . This proves the first part. Conversely, for {displaystyle varepsilon >0} , {estilo de exibição |c_{n}|geq (t-varepsilon )^{n}} for infinitely many {estilo de exibição c_{n}} , so if {estilo de exibição |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 {alfa de estilo de exibição } be a multi-index (a n-tuple of integers) com {estilo de exibição |alfa |=alfa_{1}+cdots +alfa _{n}} , então {estilo de exibição f(x)} converges with radius of convergence {estilo de exibição rho } (which is also a multi-index) se e apenas se {displaystyle limsup _{|alfa |to infty }{quadrado[{|alfa |}]{|c_{alfa }|oh ^{alfa }}}=1} to the multidimensional power series {soma de estilo de exibição _{alpha geq 0}c_{alfa }(z-a)^{alfa }:=soma _{alfa _{1}geq 0,ldots ,alfa _{n}geq 0}c_{alfa _{1},ldots ,alfa _{n}}(z_{1}-uma_{1})^{alfa _{1}}cdots (z_{n}-uma_{n})^{alfa _{n}}} A prova pode ser encontrada em [6] Notes ^ Cauchy, UMA. eu. (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, 4e 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. ^ Idioma, Sarja (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, Sociedade Americana de Matemática, ISBN 978-0821819753 Weissstein esquerdo externo, Eric W. "Cauchy-Hadamard theorem". MathWorld. Categorias: Augustin-Louis CauchyMathematical seriesTheorems in complex analysis