Possible definitions of exponential function

I was wondering how many definitions of exponential functions can we think of. The basic ones could be:

$$e^x:=\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ also $$e^x:=\lim_{n\to\infty}\bigg(1+\frac{x}{n}\bigg)^n$$ or this one: Define $e^x:\mathbb{R}\rightarrow\mathbb{R}\\$ as unique function satisfying: \begin{align} e^x\geq x+1\\ \forall x,y\in\mathbb{R}:e^{x+y}=e^xe^y \end{align} Can anyone come up with something unusual? (Possibly with some explanation or references).

Solution 1:

The exponential function is the unique solution of the initial value problem

$y'(x)=y(x) , \quad y(0)=1$.

Solution 2:

We can also define $e^x$ as follows:

• the inverse function of $\ln x$, defining $\ln x$ independently as follows

$$\ln x := \int_1^x \frac{dt}{t}$$

• the unique solution to IVP $f'(x)=f(x)$ with $f(0)=1$ which existence is guaranteed by Existence and Uniqueness” Theorems for first order IVP

Solution 3:

Define the value at rationals via powers and roots and then show that there is a unique continuous function which agrees with these values.

First define it for the natural numbers:

Define $e^2 = e \times e$, $e^3 = e \times e \times e $, etc.

Now define it for other integers:

$e^0 = 1$, $e^{-n} = \frac{1}{e^n}$, etc.

Now for other rational numbers (getting a bit harder):

$e^{\frac{p}{q}} = \sqrt[q]{e^p}$

Finally for irrational numbers $x$, you will need to prove that this definition evaluated for any sequence of rational numbers which converge to $x$ has a limit and that it is the same for all sequences which converge to $x$.

This is hard, especially the last step, but I think that it fits a common naive idea of what exponentiation is. We usually learn it in this sequence.

Solution 4:

Throw $n$ balls into $n$ bins uniformly at random, and take $n \to \infty$. Define $\frac{1}{e}$ to be the limiting fraction of empty bins.

A vehicle moves from point $A$ to $B$ with speed always equal to the remaining distance to $B$. Define $1-\frac{1}{e}$ to be the fraction of distance covered after one unit of time.

Given positive $x$, consider a set of independent Bernoulli random variables with $\sum_{i=1}^n p_i = x$. As $n \to \infty$ and $\max_i p_i \to 0$, define $e^{-x}$ to be the probability that all are zero.