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 of an analytic complex-valued function in a region of the complex plane with a continuous complex-valued function of bounded variation on the closed interval 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 is a complex-valued function of bounded variation but not necessarily continuous on a closed interval of the real axis, is a complex number not in the range of , and is a complex-valued function on such that for in . We present conditions for to be of bounded variation on .

**[1]**Gordon Thomas Whyburn,*Topological analysis*, Princeton Mathematical Series. No. 23, Princeton University Press, Princeton, N. J., 1958. MR**0099642****[2]**Fred M. Wright and Anastasios Andronikou,*The Weierstrass integral in the complex plane*, Bull. Soc. Math. Grèce (N.S.)**12**(1971), no. 1, 170–190. MR**0300183****[3]**George M. Ewing,*Calculus of variations with applications*, W. W. Norton & Co. Inc., New York, 1969. MR**0242032**

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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