Characterizations of the exponential distribution based on certain properties of its characteristic function

Next to the normal distribution, the exponential distribution is possibly the most widely referenced continuous probability law. It appears as a textbook or an inclass example in introductory probability and statistics courses, and constitutes the A and B of reliability and life testing. There exist numerous characterizations of the exponential distribution, most of them based on the "lack of memory" and the "constant hazard rate" properties. The lack of memory property states that the exponential is the only law satisfying, F(x + y) = F(x)F(y), for all x, y > 0, where