Exponential Function

Prerequisites: Calculus.

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

The exponential function describes growth in which the rate of growth is proportional to how much there already is. A bank account with interest, a population of bacteria with plenty of food, and a chain reaction all behave this way at first: twice as much stuff grows twice as fast.

The number e, about 2.718, appears when this growth happens continuously instead of in steps. Suppose a bank pays 100 percent interest per year. Paid once at the end of the year, one euro becomes two. Paid as 50 percent twice a year, the interest earned in the first half already earns interest in the second half, and one euro becomes 2.25. Splitting the year into more and more pieces gives a little more each time, but the amount does not grow without limit. It approaches e euros. The exponential function gives, for any time, what one unit becomes after growing continuously at rate 1 for that time.

The same function describes decay when the rate is negative. A radioactive sample loses a fixed fraction of its atoms per second, so the number of atoms falls by the same factor in every equal time interval. That is why every radioactive isotope has a fixed half-life.

In physics the exponential turns up wherever a rate of change is proportional to the current value. With an imaginary rate it describes rotation and oscillation instead of growth (see Euler's formula), which is why it appears in almost every equation of wave physics and quantum mechanics.

Concrete

The exponential function can be defined in three equivalent ways:

$$e^x = \lim_{n \to \infty}\left(1 + \frac{x}{n}\right)^n = \sum_{k = 0}^{\infty} \frac{x^k}{k!},$$

or as the unique solution of

$$\frac{d f}{dx} = f, \qquad f(0) = 1.$$

The number $e = e^1 = \sum_k 1/k! \approx 2.71828$. Differentiating the series term by term shows directly that it equals its own derivative. The function satisfies $e^{x + y} = e^x e^y$, $e^0 = 1$ and $e^x > 0$ for all real $x$. Its inverse is the natural logarithm $\ln$, and for any base $a > 0$ we have $a^x = e^{x \ln a}$.

Growth and decay. The equation $dN/dt = kN$ has the solution $N(t) = N_0\, e^{kt}$, since $\frac{d}{dt} N_0 e^{kt} = k N_0 e^{kt}$. For radioactive decay $k = -\lambda < 0$. The half-life $t_{1/2}$ is defined by $N(t_{1/2}) = N_0/2$, so $e^{-\lambda t_{1/2}} = 1/2$ and

$$t_{1/2} = \frac{\ln 2}{\lambda}.$$

The mean lifetime of a nucleus is $\tau = 1/\lambda$, after which a fraction $1/e \approx 0.37$ of the sample is left.

Boltzmann factor. In statistical mechanics the probability of finding a system in equilibrium at temperature $T$ in a state of energy $E$ is proportional to $e^{-E/k_B T}$. Exponentials also appear as solutions of every linear differential equation with constant coefficients: the ansatz $x(t) = e^{rt}$ turns $\ddot x + a\dot x + bx = 0$ into the algebraic equation $r^2 + ar + b = 0$.

Complex argument. The series converges for every complex $z$, and with $z = i\theta$ it gives Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$. Plane waves $e^{i(kx - \omega t)}$ and the time evolution $e^{-iEt/\hbar}$ of an energy eigenstate in quantum mechanics are of this form.

• The best introduction is An Intuitive Guide To Exponential Functions & e by Kalid Azad
• See also How To Think With Exponents And Logarithms by Kalid Azad

Abstract

The continuous functions $f: \mathbb{R} \to \mathbb{R}$ with $f(x + y) = f(x) f(y)$ that are not identically zero are exactly $x \mapsto e^{cx}$, $c \in \mathbb{R}$. They are the continuous group homomorphisms from $(\mathbb{R}, +)$ to $(\mathbb{R}_{>0}, \cdot)$, and $\exp$ is an isomorphism of these two groups.

The power series makes sense in any Banach algebra $\mathcal{A}$ (for example the $n \times n$ matrices, or the bounded operators on a Hilbert space):

$$e^{A} = \sum_{k = 0}^\infty \frac{A^k}{k!},$$

which converges absolutely since $\|A^k\| \leq \|A\|^k$. If $AB = BA$ then $e^{A + B} = e^A e^B$. In general this fails, and the Baker-Campbell-Hausdorff formula gives $e^A e^B = e^{C}$ with

$$C = A + B + \tfrac{1}{2}[A, B] + \tfrac{1}{12}\big([A, [A, B]] + [B, [B, A]]\big) + \dots$$

for $A, B$ small enough, where the dots are higher nested commutators. The map $t \mapsto e^{tA}$ is a one-parameter group, $e^{(s + t)A} = e^{sA}e^{tA}$, and solves $\dot X = AX$ with $X(0) = \mathbb{1}$.

For a matrix Lie group $G \subset GL(n, \mathbb{C})$ with Lie algebra $\mathfrak{g}$, the matrix exponential restricts to the exponential map $\exp: \mathfrak{g} \to G$. It is a local diffeomorphism near $0$, and its image generates the connected component of the identity. It need not be surjective: in $SL(2, \mathbb{R})$ the element $\mathrm{diag}(-2, -1/2)$ is not the exponential of any traceless real matrix.

In quantum mechanics the generators are unbounded, and the series no longer applies directly. Stone's theorem gives the replacement: every strongly continuous one-parameter group $U(t)$ of unitary operators on a Hilbert space has the form $U(t) = e^{-itH}$ for a unique self-adjoint (possibly unbounded) operator $H$, defined through the spectral theorem as $e^{-itH} = \int e^{-it\lambda}\, dP(\lambda)$, and conversely. With $H$ the Hamiltonian divided by $\hbar$, this is the time evolution operator.

Why is it interesting?

Describing e as “a constant approximately 2.71828…” is like calling pi “an irrational number, approximately equal to 3.1415…”. Sure, it’s true, but you completely missed the point.

Pi is the ratio between circumference and diameter shared by all circles. It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface area for circles, spheres, cylinders, and so on. Pi is important and shows all circles are related, not to mention the trigonometric functions derived from circles (sin, cos, tan).

e is the base rate of growth shared by all continually growing processes. e lets you take a simple growth rate (where all change happens at the end of the year) and find the impact of compound, continuous growth, where every nanosecond (or faster) you are growing just a little bit.

e shows up whenever systems grow exponentially and continuously: population, radioactive decay, interest calculations, and more. Even jagged systems that don’t grow smoothly can be approximated by e.

Just like every number can be considered a scaled version of 1 (the base unit), every circle can be considered a scaled version of the unit circle (radius 1), and every rate of growth can be considered a scaled version of e (unit growth, perfectly compounded).

So e is not an obscure, seemingly random number. e represents the idea that all continually growing systems are scaled versions of a common rate. An Intuitive Guide To Exponential Functions & e by Kalid Azad

Leads to: Euler's Formula, Logarithm.