Functional equations with trigonometric function solutions

Abstract

The first part of the thesis treats the problem of characterizing the trigonometric and hyperbolic sine-cosine functions. Our method arises from Kannappan's work of 2003 which solved the functional equation f(x - y) = f(x)f(y) + g(x)g(y) for functions whose domain is a group, whose range is a subset of the complex field and without any additional conditions. We use Kannapan's technique to determine the general solutions of the functional equation f(x+y) = f(x)f(y) - g(x)g(y) which together with Kannappan's result give a complete characterization of the trigonometric sine-cosine functions. Next, the functional equation f(x - y) = f(x)f(y) - g(x)g(y) is used to characterize the hyperbolic sine-cosine functions, and inter-relations among the solution functions, resemble certain well-known hyperbolic sine-cosine identities and generalizing the classical d'Alembert functional equation, are obtained. The second part of the thesis gives characterizations of the trigonometric and hyperbolic tangent-cotangent functions. There are two approaches in this part. The first approach is along the line treated by Dobbs in 1989 for the trigonometric tangent function. It is analytic in character and makes use of continuity and differentiabilty at one specific point. Dobbs defined the class of real-valued functions T of real variable, called tangential functions, as those satisfying the functional equation T(u + v) = [T(u)+T(v)]/[1-T(u)T(v)]. We apply the result of Dobbs to characterize the trigonometric cotangent function and then proceed to use Dobbs' approach to characterize the hyperbolic tangent-cotangent functions through their respective functional equations. The functions considered are to have the real numbers and/or its subset as their domain and range. The second approach is discrete in character and stems from the work of Rhouma in 2005 which gave a closed form solution to the recursive difference equation yn+2=(ynyn+1-1)/(yn+ yn+1) . This is a discrete functional equation of much recent interests in itself. We generalize the technique of Rhouma to find the closed form solutions of certain rational recursive equations and use the results to characterize the cotangent-tangent and the hyperbolic cotangent-tangent functions.