Characterizations of the exponential function

In mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent of and equivalent to each other. As a special case of these considerations, we will see that the three most common definitions given for the mathematical constant e are also equivalent to each other.

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Characterizations

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

1. Define ex by the limit
e^x = \lim_{n\to\infty} \left(1%2B\frac{x}{n}\right)^n.
2. Define ex as the value of the infinite series
e^x = \sum_{n=0}^\infty {x^n \over n!} = 1 %2B x %2B \frac{x^2}{2!} %2B \frac{x^3}{3!} %2B \frac{x^4}{4!} %2B \cdots
(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
\int_{1}^{y} \frac{dt}{t} = x.
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
y'=y,\quad y(0)=1.
5. The exponential function f(x) = ex is the unique Lebesgue-measurable function with f(1) = e that satisfies
f(x%2By) = f(x) f(y)\text{ for all }x\text{ and }y \,
(Hewitt and Stromberg, 1965, exercise 18.46). Alternatively, it is the unique anywhere-continuous function with these properties (Rudin, 1976, chapter 8, exercise 6).
(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.

Larger domains

One way of defining the exponential function for domains larger than the domain of real numbers is to first define it for the domain of real numbers using one of the above characterizations and then extend it to larger domains in a way which would work for any analytic function.

It is also possible to use the characterisations directly for the larger domain, though some problems may arise. (1), (2), and (4) all make sense for arbitrary Banach algebras. (3) presents a problem for complex numbers, because there are non-equivalent paths along which one could integrate, and (5) is not sufficient. For example, the function f defined (for x and y real) as

f(x %2B iy) = e^x(\cos(2y) %2B i\sin(2y)) = e^{x %2B 2iy} \,

satisfies the conditions in (5) without being the exponential function of x + iy. To make (5) sufficient for the domain of complex numbers, one may either stipulate that there exists a point at which f is a conformal map or else stipulate that

f(i) = \cos(1) %2B i\sin(1). \,

Why each characterization makes sense

Each characterization requires some justification to show that it makes sense. For instance, when the value of the function is defined by a sequence or series, the convergence of this sequence or series needs to be established.

Characterization 2

Since

\lim_{n\to\infty} \left|\frac{x^{n%2B1}/(n%2B1)!}{x^n/n!}\right| = \lim_{n\to\infty} \left|\frac{x}{n%2B1}\right| = 0 < 1 \mbox{.}

it follows from the ratio test that \sum_{n=0}^{\infty} \frac{x^n}{n!} converges for all x.

Characterization 3

Since the integrand is an integrable function of t, the integral expression makes sense. That every real number x corresponds to a unique y > 0 such that

\int_1^y \frac{dt}{t} = x

is equivalent to the statement that the integral is a bijection from the interval (0,%2B\infty) to (-\infty,%2B\infty), which follows if one can show that 1/t is positive for positive t (so the function is monotone increasing, hence one-to-one) and that the two integrals

\int_1^\infty \frac{dt}{t} = \infty
\int_0^1 \frac{dt}{t} = -\infty

hold, so it is onto.

The first statement is obvious – t>0 implies 1/t > 0 – and the two integrals follow from the integral test and the divergence of the harmonic series.

Equivalence of the characterizations

The following proof demonstrates the equivalence of the three characterizations given for e above. The proof consists of two parts. First, the equivalence of characterizations 1 and 2 is established, and then the equivalence of characterizations 1 and 3 is established.

Equivalence of characterizations 1 and 2

The following argument is adapted from a proof in Rudin, theorem 3.31, p. 63-65.

Let x\geq0 be a fixed non-negative real number. Define

s_n = \sum_{k=0}^n\frac{x^k}{k!},\ t_n=\left(1%2B\frac{x}{n}\right)^n.

By the binomial theorem,

\begin{align} t_n & =\sum_{k=0}^n{n \choose k}\frac{x^k}{n^k}=1%2Bx%2B\sum_{k=2}^n\frac{n(n-1)(n-2)\cdots(n-(k-1))x^k}{k!\,n^k} \\[8pt] & = 1%2Bx%2B\frac{x^2}{2!}\left(1-\frac{1}{n}\right)%2B\frac{x^3}{3!}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)%2B\cdots \\[8pt] & {}\qquad \cdots %2B\frac{x^n}{n!}\left(1-\frac{1}{n}\right)\cdots\left(1-\frac{n-1}{n}\right)\le s_n \end{align}

(using x ≥ 0 to obtain the final inequality) so that

\limsup_{n\to\infty}t_n \le \limsup_{n\to\infty}s_n = e^x

where ex is in the sense of definition 2. Here, we must use limsups, because we don't yet know that tn actually converges. Now, for the other direction, note that by the above expression of tn, if 2 ≤ m ≤ n, we have

1%2Bx%2B\frac{x^2}{2!}\left(1-\frac{1}{n}\right)%2B\cdots%2B\frac{x^m}{m!}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)\cdots\left(1-\frac{m-1}{n}\right)\le t_n.

Fix m, and let n approach infinity. We get

s_m = 1%2Bx%2B\frac{x^2}{2!}%2B\cdots%2B\frac{x^m}{m!} \le \liminf_{n\to\infty}t_n

(again, we must use liminf's because we don't yet know that tn converges). Now, take the above inequality, let m approach infinity, and put it together with the other inequality. This becomes

\limsup_{n\to\infty}t_n \le e^x \le \liminf_{n\to\infty}t_n \,

so that

\lim_{n\to\infty}t_n = e^x. \,

We can then extend this equivalence to the negative real numbers by noting \left(1-\frac{r}{n}\right)^n \left(1%2B\frac{r}{n}\right)^n = \left(1-\frac{r^2}{n^2}\right)^n and taking the limit as n goes to infinity.

The error term of this limit-expression is described by

\left(1%2B\frac{x}{n}\right)^n=e^x \left(1-\frac{x^2}{2n}%2B\frac{x^3(8%2B3x)}{24n^2}%2B\cdots \right),

where the polynomial's degree (in x) in the term with denominator nk is 2k.

Equivalence of characterizations 1 and 3

Here, we define the natural logarithm function in terms of a definite integral as above. By the fundamental theorem of calculus,

\frac{d}{dx}\left( \ln x \right) = \frac{1}{x}.

Now, let x be any fixed real number, and let

y=\lim_{n\to\infty}\left(1%2B\frac{x}{n}\right)^n.

We will show that ln(y) = x, which implies that y = ex, where ex is in the sense of definition 3. We have

\ln y=\ln\lim_{n\to\infty}\left(1%2B\frac{x}{n}\right)^n=\lim_{n\to\infty}\ln\left(1%2B\frac{x}{n}\right)^n.

Here, we have used the continuity of ln(y), which follows from the continuity of 1/t:

\ln y=\lim_{n\to\infty}n\ln \left(1%2B\frac{x}{n}\right)=\lim_{n\to\infty}\frac{x\ln\left(1%2B(x/n)\right)}{(x/n)}.

Here, we have used the result lnan = nlna. This result can be established for n a natural number by induction, or using integration by substitution. (The extension to real powers must wait until ln and exp have been established as inverses of each other, so that ab can be defined for real b as eb lna.)

=x\cdot\lim_{h\to 0}\frac{\ln\left(1%2Bh\right)}{h} \quad \mbox{ where }h=\frac{x}{n}
=x\cdot\frac{d}{dt}\left( \ln t\right) \quad \mbox{ at }t=1
=x\cdot\frac{1}{t} \quad \mbox{ at }t=1
\!\, = x.

Equivalence of characterizations 1 and 5

The following proof is a simplified version of the one in Hewitt and Stromberg, exercise 18.46. First, one proves that measurability (or here, Lebesgue-integrability) implies continuity for a non-zero function f(x) satisfying f(x%2By)=f(x)f(y), and then one proves that continuity implies f(x) = e^{kx} for some k, and finally f(1)=e implies k=1.

First, we prove a few elementary properties from f(x) satisfying f(x%2By)=f(x)f(y) and the assumption that f(x) is not identically zero:

The second and third properties mean that it is sufficient to prove f(x)=e^x for positive x.

If f(x) is a Lebesgue-integrable function, then we can define

g(x) = \int_0^x f(x')\, dx'.

It then follows that

g(x%2By)-g(x) = \int_x^{x%2By} f(x')\, dx' = \int_0^y f(x%2Bx')\, dx' = f(x)g(y).

Since f(x) is nonzero, we can choose some y such that g(y) \neq 0 and solve for f(x) in the above expression. Therefore:

f(x%2B\delta)-f(x) = \frac{[g(x%2B\delta%2By)-g(x%2B\delta)]-[g(x%2By)-g(x)]}{g(y)}
=\frac{[g(x%2By%2B\delta)-g(x%2By)]-[g(x%2B\delta)-g(x)]}{g(y)}
=\frac{f(x%2By)g(\delta)-f(x)g(\delta)}{g(y)}=g(\delta)\frac{f(x%2By)-f(x)}{g(y)}.

The final expression must go to zero as \delta\rightarrow 0 since g(0)=0 and g(x) is continuous. It follows that f(x) is continuous.

Now, we prove that f(q) = e^{kq}, for some k, for all positive rational numbers q. Let q=n/m for positive integers n and m. Then

f\left(\frac{n}{m}\right)=f\left(\frac{1}{m}%2B\cdots%2B\frac{1}{m}\right)=f\left(\frac{1}{m}\right)^{n}

by elementary induction on n. Therefore, f(1/m)^{m}=f(1) and thus

f\left(\frac{n}{m}\right)=f(1)^{n/m}=e^{k(n/m)}.

for k = \ln [f(1)]\,. Note that if we are restricting ourselves to real-valued f(x), then f(x) = f(x/2)^2 is everywhere positive and so k is real.

Finally, by continuity, since f(x) = e^{kx} for all rational x, it must be true for all real x since the closure of the rationals is the reals (that is, we can write any real x as the limit of a sequence of rationals). If f(1) = e then k = 1. This is equivalent to characterization 1 (or 2, or 3), depending on which equivalent definition of e one uses.

References