Cauchy–Hadamard theorem

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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]

Theory for one complex variable[edit]

Statement of the theorem[edit]

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 of ƒ at the point a is given by

\frac{1}{R} = \limsup_{n \to \infty} \big( | c_{n} |^{1/n} \big)

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 of the theorem[edit]

[5] Without loss of generality assume that a=0. We will show first that the power series \sum 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 zero or ±infinity. For any \epsilon > 0, there exists only a finite number of n such that
\sqrt[n]{|c_n|}\geq t+\epsilon. Now |c_n|\leq(t+\epsilon)^n for all but a finite number of c_n, so the series \sum c_n z^n converges if |z| < 1/(t+\epsilon). This proves the first part.

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

The case where t is zero or ±infinity is left for the reader.

Several complex variables[edit]

Statement of the theorem[edit]

Let \alpha be a multi-index (a n-tuple of integers) with |\alpha|=\alpha_1+\ldots+\alpha_n, then f(x) converges with radius of convergence \rho (which is also a multi-index) if and only if

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

to the multidimensional power series

\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}\ldots(z_n-a_n)^{\alpha_n}

Proof of the theorem[edit]

The proof can be found in the book Introduction to Complex Analysis Part II functions in several Variables by B.V.Shabat

Notes[edit]

  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, 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.
  5. ^ Lang, Serge (2002), Complex Analysis: Fourth Edition, Springer, pp. 55–56, ISBN 0-387-98592-1 Graduate Texts in Mathematics

External links[edit]