Characterizations of the exponential function

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the exponential function

Exponential function

In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

can be characterized

Characterization (mathematics)

In mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in...

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

E (mathematical constant)

The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

are also equivalent to each other.

Characterizations

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

1. Define e^x by the limit

Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

2. Define e^x as the value of the infinite series

(Here n! stands for the factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

of n. One proof that e is irrational

Proof that e is irrational

In mathematics, the series representation of Euler's number ecan be used to prove that e is irrational. Of the many representations of e, this is the Taylor series for the exponential function evaluated at y = 1.-Summary of the proof:...

uses this representation.)

3. Define e^x to be the unique number y > 0 such that

This is as the inverse of the natural logarithm

Natural logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

function, which is defined by this integral.

4. Define e^x to be the unique solution to the initial value problem

Initial value problem

In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...

5. The exponential function f(x) = e^x is the unique Lebesgue-measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...

with f(1) = e that satisfies

(Hewitt and Stromberg, 1965, exercise 18.46). Alternatively, it is the unique anywhere-continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

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) = e^x is guaranteed for rational x by the above properties (see below), one could also use monotonicity or other properties to enforce the choice of e^x 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

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

.

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 algebra

Banach algebra

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...

s. (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

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

Conformal map

In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

or else stipulate that

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

it follows from the ratio test that

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

is equivalent to the statement that the integral is a bijection

Bijection

A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

from the interval

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

hold, so it is onto.

The first statement is obvious –

implies

– and the two integrals follow from the integral test and the divergence of the harmonic series

Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...

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

be a fixed non-negative real number. Define

By the binomial theorem

Binomial theorem

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with , and the coefficient a of...

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

where e^x is in the sense of definition 2. Here, we must use limsups

Limit superior and limit inferior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence...

, because we don't yet know that t_n actually converges

Limit (mathematics)

. Now, for the other direction, note that by the above expression of t_n, if 2 ≤ m ≤ n, we have

Fix m, and let n approach infinity. We get

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

so that

We can then extend this equivalence to the negative real numbers by noting

and taking the limit as n goes to infinity.

The error term of this limit-expression is described by

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

Equivalence of characterizations 1 and 3

Here, we define the natural logarithm

Natural logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

function in terms of a definite integral as above. By the fundamental theorem of calculus

Fundamental theorem of calculus

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

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

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

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

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 a^b can be defined for real b as e^{b lna}.)

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

satisfying

, and then one proves that continuity implies

for some k, and finally

implies k=1.

First, we prove a few elementary properties from

satisfying

and the assumption that

is not identically zero:

If is nonzero anywhere (say at x=y), then it is non-zero everywhere. Proof: implies .
. Proof: and is non-zero.
. Proof: .
If is continuous anywhere (say at x=y), then it is continuous everywhere. Proof: as by continuity at y.

The second and third properties mean that it is sufficient to prove

for positive x.

If

is a Lebesgue-integrable function, then we can define

It then follows that

Since

is nonzero, we can choose some y such that

and solve for

in the above expression. Therefore:

The final expression must go to zero as

since

and

is continuous. It follows that

is continuous.

Now, we prove that

, for some k, for all positive rational numbers q. Let q=n/m for positive integers n and m. Then

by elementary induction on n. Therefore,

and thus

for

. Note that if we are restricting ourselves to real-valued

, then

is everywhere positive and so k is real.

Finally, by continuity, since

for all rational x, it must be true for all real x since the closure

Closure (mathematics)

In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

of the rationals is the reals (that is, we can write any real x as the limit of a sequence of rationals). If

then k = 1. This is equivalent to characterization 1 (or 2, or 3), depending on which equivalent definition of e

E (mathematical constant)

one uses.

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