The exponential function occurs naturally in many branches of mathematics. Walter Rudin called it "the most important function in mathematics".[1]
It is therefore useful to have multiple ways to define (or characterize) it.
Each of the characterizations below may be more or less useful depending on context.
The "product limit" characterization of the exponential function was discovered by Leonhard Euler.[2]
Characterizations
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The six most common definitions of the exponential function for real values are as follows.
Inverse of logarithm integral. Define to be the unique number y > 0 such that That is, is the inverse of the natural logarithm function , which is defined by this integral.
Functional equation. The exponential function is the unique function f with the multiplicative property for all and . The condition can be replaced with together with any of the following regularity conditions:
For the uniqueness, one must impose some regularity condition, since other functions satisfying can be constructed using a basis for the real numbers over the rationals, as described by Hewitt and Stromberg.
Elementary definition by powers. Define the exponential function with base to be the continuous function whose value on integers is given by repeated multiplication or division of , and whose value on rational numbers is given by . Then define to be the exponential function whose base is the unique positive real number satisfying:
Larger domains
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One way of defining the exponential function over the complex numbers is to first define it for the domain of real numbers using one of the above characterizations, and then extend it as an analytic function, which is characterized by its values on any infinite domain set.
Also, characterisations (1), (2), and (4) for apply directly for a complex number. Definition (3) presents a problem because there are non-equivalent paths along which one could integrate; but the equation of (3) should hold for any such path modulo . As for definition (5), the additive property together with the complex derivative are sufficient to guarantee . However, the initial value condition together with the other regularity conditions are not sufficient. For example, for real x and y, the functionsatisfies the three listed regularity conditions in (5) but is not equal to . A sufficient condition is that and that is a conformal map at some point; or else the two initial values and together with the other regularity conditions.
One may also define the exponential on other domains, such as matrices and other algebras. Definitions (1), (2), and (4) all make sense for arbitrary Banach algebras.
Proof that each characterization makes sense
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Some of these definitions require justification to demonstrate that they are well-defined. For example, when the value of the function is defined as the result of a limiting process (i.e. an infinite sequence or series), it must be demonstrated that such a limit always exists.
Characterization 1
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The error of the product limit expression is described by:
where the polynomial's degree (in x) in the term with denominator nk is 2k.
Characterization 2
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Since
it follows from the ratio test that converges for all x.
Characterization 3
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Since the integrand is an integrable function of t, the integral expression is well-defined. It must be shown that the function from to defined by
is a bijection. Since 1/t is positive for positive t, this function is strictly increasing, hence injective. If the two integrals
hold, then it is surjective as well. Indeed, these integrals do hold; they follow from the integral test and the divergence of the harmonic series.
Characterization 6
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The definition depends on the unique positive real number satisfying: This limit can be shown to exist for any , and it defines a continuous increasing function with and , so the Intermediate value theorem guarantees the existence of such a value .
Equivalence of the characterizations
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The following arguments demonstrate the equivalence of the above characterizations for the exponential function.
Characterization 1 ⇔ characterization 2
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The following argument is adapted from Rudin, theorem 3.31, p. 63–65.
Let be a fixed non-negative real number. Define
By the binomial theorem,
(using x ≥ 0 to obtain the final inequality) so that:
One must use lim sup because it is not known if tnconverges.
For the other inequality, by the above expression for tn, if 2 ≤ m ≤ n, we have:
Fix m, and let n approach infinity. Then
(again, one must use lim inf because it is not known if tn converges). Now, take the above inequality, let m approach infinity, and put it together with the other inequality to obtain:
so that
This equivalence can be extended to the negative real numbers by noting and taking the limit as n goes to infinity.
Ln(y) = x, which implies that y = ex, where ex is in the sense of definition 3. We have
Here, the continuity of ln(y) is used, which follows from the continuity of 1/t:
Here, the result lnan = nlna has been used. 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.)
Characterization 1 ⇔ characterization 4
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Let denote the solution to the initial value problem . Applying the simplest form of Euler's method with increment and sample points gives the recursive formula:
This recursion is immediately solved to give the approximate value , and since Euler's Method is known to converge to the exact solution, we have:
Characterization 2 ⇔ characterization 4
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Let n be a non-negative integer. In the sense of definition 4 and by induction, .
Therefore
Using Taylor series,
This shows that definition 4 implies definition 2.
In the sense of definition 2,
Besides, This shows that definition 2 implies definition 4.
Characterization 2 ⇒ characterization 5
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In the sense of definition 2, the equation follows from the term-by-term manipulation of power series justified by uniform convergence, and the resulting equality of coefficients is just the Binomial theorem. Furthermore:[3]
Characterization 3 ⇔ characterization 4
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Characterisation 3 first defines the natural logarithm:then as the inverse function with . Then by the Chain rule:
i.e. . Finally, , so . That is, is the unique solution of the initial value problem , of characterization 4.
Conversely, assume has and , and define as its inverse function with and . Then:
The conditions f'(0) = 1 and f(x + y) = f(x) f(y) imply both conditions in characterization 4. Indeed, one gets the initial condition f(0) = 1 by dividing both sides of the equation
by f(0), and the condition that f′(x) = f(x) follows from the condition that f′(0) = 1 and the definition of the derivative as follows:
Characterization 5 ⇒ characterization 4
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Assum characterization 5, the multiplicative property together with the initial condition imply that:
Characterization 5 ⇔ characterization 6
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By inductively applying the multiplication rule, we get:
and thus
for . Then the condition means that , so by definition.
Also, any of the regularity conditions of definition 5 imply that is continuous at all real (see below). The converse is similar.
Characterization 5 ⇒ characterization 6
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Let be a Lebesgue-integrable non-zero function satisfying the mulitiplicative property with . Following Hewitt and Stromberg, exercise 18.46, we will prove that Lebesgue-integrability implies continuity. This is sufficient to imply according to characterization 6, arguing as above.
First, a few elementary properties:
If is nonzero anywhere (say at ), then it is non-zero everywhere. Proof: implies .
. Proof: and is non-zero.
. Proof: .
If is continuous anywhere (say at ), then it is continuous everywhere. Proof: as by continuity at .
The second and third properties mean that it is sufficient to prove for positive x.