Admissible exponential representations and topological indices for functions of bounded variation

Abstract

In this paper we first prove a theorem concerning the composition η \eta of an analytic complex-valued function g g in a region of the complex plane with a continuous complex-valued function ϕ \phi of bounded variation on the closed interval [ a , b ] [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 ϕ \phi is a complex-valued function of bounded variation but not necessarily continuous on a closed interval [ a , b ] [a,b] of the real axis, p p is a complex number not in the range of ϕ \phi , and u u is a complex-valued function on [ a , b ] [a,b] such that e u ( t ) = [ ϕ ( t ) − p ] {e^{u(t)}} = [\phi (t) - p] for t t in [ a , b ] [a,b] . We present conditions for u u to be of bounded variation on [ a , b ] [a,b] .