# Cauchy-Hadamard Theorem

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## Cauchy-Hadamard theorem

[kō·shē ′had·ə·mär ‚thir·əm] (mathematics)

The theorem that the radius of convergence of a Taylor series in the complex variable

*z*is the reciprocal of the limit superior, as*n*approaches infinity, of the*n*th root of the absolute value of the coefficient of*z*.^{n}McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

The following article is from

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.## Cauchy-Hadamard Theorem

a theorem of the theory of analytic functions that permits the evaluation of the convergence of the power series

a_{0} + a_{1}(z − z_{0}) + … a_{n}(z − z_{0})^{n} + …

where a_{0}, a_{1}, …, a_{n} are fixed complex numbers and z is a complex variable. The Cauchy-Hadamard theorem states that if the upper limit

then when ρ = ∞ the series converges absolutely in the entire plane. When ρ = 0 the series converges only at the point z = z_{0} and diverges when z ≠ z_{0}. Finally, for the case 0 < ρ < ∞, the series converges absolutely in the circle ǀ z − Z_{0}ǀ< ρ and diverges outside it. The theorem was established by A. Cauchy (1821), and a second proof of it was given by J. Hadamard (1888), who indicated its important applications.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.