Characterizations of The Exponential Function - Characterizations

Characterizations

The five most common definitions of the exponential function exp(x) = ex for real x are:

1. Define ex by the limit

2. Define ex as the value of the infinite series

(Here n! stands for the factorial of n. One proof that e is irrational uses this representation.)

3. Define ex to be the unique number y > 0 such that

This is as the inverse of the natural logarithm function, which is defined by this integral.

4. Define ex to be the unique solution to the initial value problem

(Here y' stands for the derivative of y.)

5. The exponential function f(x) = ex is the unique Lebesgue-measurable function with f(1) = e that satisfies

(Hewitt and Stromberg, 1965, exercise 18.46). Alternatively, it is the unique anywhere-continuous function with these properties (Rudin, 1976, chapter 8, exercise 6). The term "anywhere-continuous" means that there exists at least a single point at which is continuous. As shown below, if for all and and is continuous at any single point then is necessarily continuous everywhere.

(As a counterexample, if one does not assume continuity or measurability, it is possible to prove the existence of an everywhere-discontinuous, non-measurable function with this property by using a Hamel basis for the real numbers over the rationals, as described in Hewitt and Stromberg.)

Because f(x) = ex is guaranteed for rational x by the above properties (see below), one could also use monotonicity or other properties to enforce the choice of ex for irrational x, but such alternatives appear to be uncommon.