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] Contents 1 Theorem for one complex variable 1.1 Proof 2 Theorem for several complex variables 3 Notes 4 External links 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 f at the point a is given by {displaystyle {frac {1}{R}}=limsup _{nto 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 {displaystyle a=0} . We will show first that the power series {textstyle sum _{n}c_{n}z^{n}} converges for {displaystyle |z|R} .

First suppose {displaystyle |z|0} , there exists only a finite number of {displaystyle n} such that {textstyle {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 {textstyle sum _{n}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 limsup _{|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 ^ 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, 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. ^ 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, ISBN 978-0821819753 External links Weisstein, Eric W. "Cauchy-Hadamard theorem". MathWorld. Categories: Augustin-Louis CauchyMathematical seriesTheorems in complex analysis