In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes sense, and proves that they are all equivalent.

The exponential function occurs naturally in many branches of mathematics. Walter Rudin called it "the most important function in mathematics". 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.

Characterizations

are as follows.

1. Product limit. by the limit

2. Power series. Define as the value of the infinite series (Here denotes the factorial of . One proof that e is irrational uses a special case of this formula.)

3. Inverse of logarithm integral. to be the unique number is the inverse of the natural logarithm, which is defined by this integral.

4. Differential equation. to be the unique solution to the differential equation with initial value denotes the derivative of .

5. Functional equation. is the unique function together with any of the following regularity conditions: For the uniqueness, one must impose some can be constructed using a basis for the real numbers over the rationals, as described by Hewitt and Stromberg.

6. Elementary definition by powers.

Larger domains

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.

a complex number. As for definition (5), the additive property together with the complex derivative together with the other regularity conditions are not sufficient. For example, for real x and y is a conformal map 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

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

where the polynomial's degree (in ) in the term with denominator is.

Characterization 2

it follows from the ratio test converges for all .

Characterization 3

Since the integrand is an integrable function of is a bijection. Since is positive for positive , this function is strictly increasing, hence injective 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

, so the Intermediate value theorem.

Equivalence of the characterizations

The following arguments demonstrate the equivalence of the above characterizations for the exponential function.

Characterization 1 ⇔ characterization 2

The following argument is adapted from Rudin, theorem 3.31, p.63–65.

By the binomial theorem (using One must use lim sup because it is not known if converges.

For the other inequality, by the above expression for , if

Fix , and let (again, one must use lim inf because it is not known if converges). Now, take the above inequality, let

and taking the limit as goes to infinity.

Characterization 1 ⇔ characterization 3

Here, the natural logarithm function is defined in terms of a definite integral as above. By the first part of fundamental theorem of calculus

Now, let

Now, , which implies that , where

Here, the continuity of is used, which follows from the continuity of

Here, the result has been used. This result can be established for 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 can be defined for real as

Characterization 1 ⇔ characterization 4

. Applying the simplest form of Euler's method

Characterization 2 ⇔ characterization 4

.

Using Taylor series This shows that definition 4 implies definition 2.

This shows that definition 2 implies definition4.

Characterization 2 ⇒ characterization 5

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:

Characterization 3 ⇔ characterization 4

. Then by the Chain rule of characterization 4.

. By the Fundamental theorem of calculus

Characterization 5 ⇒ characterization 4

The conditions and imply both conditions in characterization 4. Indeed, one gets the initial condition by , and the condition that follows from the condition that

Characterization 5 ⇒ characterization 4

Characterization 5 ⇔ characterization 6

by definition.

(see below). The converse is similar.

Characterization 5 ⇒ characterization 6

according to characterization 6, arguing as above.

First, a few elementary properties:

1. .

2. is non-zero.

3. .

4. .

for positive.

is a Lebesgue-integrable function

is nonzero, some

is continuous.

References

• Walter Rudin, Principles of Mathematical Analysis, 3rd edition (McGraw–Hill, 1976), chapter 8.

• Edwin Hewitt and Karl Stromberg, Real and Abstract Analysis (Springer, 1965).