Exponential Functional Equations

Abstract

The exponential function \(f(x)=e^x\) is a powerful tool in each field of natural sciences and engineering since many natural phenomena well-known to us can be described best of all by means of it. The famous exponential functional equation \(f(x+y)=f(x)f(y)\) simplifies the elegant property of the exponential function, for example, exCy D exey. In Section 9.1, the superstability of the exponential functional equation will be proved. Section 9.2 deals with the stability of the exponential equation in the sense of R. Ger. Stability problems of the exponential functional equation on a restricted domain and asymptotic behaviors of exponential functions are discussed in Section 9.3. Another exponential functional equation \(f(xy)=f(x)^y\) will be introduced in Section 9.4.