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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Admissible exponential representations and topological indices for functions of bounded variation


Authors: F. M. Wright and J. N. Ling
Journal: Proc. Amer. Math. Soc. 40 (1973), 431-437
MSC: Primary 30A90
MathSciNet review: 0324053
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Abstract: In this paper we first prove a theorem concerning the composition $ \eta $ of an analytic complex-valued function $ g$ in a region of the complex plane with a continuous complex-valued function $ \phi $ of bounded variation on the closed interval $ [a,b]$ of the real axis. We then relate this theorem to admissible exponential representations and topological indices introduced by Whyburn in his book Topological analysis. We also show how this theorem can be used to prove a result of interest in the study of the argument principle. Moreover, we look at the situation where $ \phi $ is a complex-valued function of bounded variation but not necessarily continuous on a closed interval $ [a,b]$ of the real axis, $ p$ is a complex number not in the range of $ \phi $, and $ u$ is a complex-valued function on $ [a,b]$ such that $ {e^{u(t)}} = [\phi (t) - p]$ for $ t$ in $ [a,b]$. We present conditions for $ u$ to be of bounded variation on $ [a,b]$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0324053-8
Keywords: Continuous function, bounded variation, admissible exponential representation, topological index, oriented Fréchet curve, argument principle, partition of a closed interval of the real axis, composite function, Stieltjes integral
Article copyright: © Copyright 1973 American Mathematical Society