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Analytic Properties of Power Product Expansions

Published online by Cambridge University Press:  20 November 2018

H. Gingold  and
A. Knopfmacher
H. Gingold
Affiliation:
Department of Mathematics West Virginia University Morgantown, West Virginia 26506 U.S.A.
A. Knopfmacher
Affiliation:
Department of Computational and Applied Mathematics University of the Witwatersrand Johannesburg, P.O. Wits 2050 South Africa e-mail: arnoldk@gauss.cam.wits.ac.za
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Abstract

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Let ƒ(z) be a complex function analytic in some neighbourhood of the origin with ƒ(0) = 1. It is known that ƒ(z) admits a unique "power product" expansion of the form convergent near zero. We derive a simple direct bound for the radius of convergence of this product expansion in terms of the coefficients of ƒ(z). In addition we show that the same bound holds in the case of "inverse power product" expansions Examples are given for which these bounds are sharp. We show also that products with nonnegative coefficients have the same radius of convergence as their corresponding series.

Keywords

41A1030E10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 6 , 01 December 1995 , pp. 1219 - 1239
Copyright
Copyright © Canadian Mathematical Society 1995

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