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".
The "product limit" characterization of the exponential function was discovered by Leonhard Euler.
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.
As for definition (5), the additive property together with the complex derivative
satisfies the three listed regularity conditions in (5) but is not equal to
is a conformal map at some point; or else the two initial values
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.
Some of these definitions require justification to demonstrate that they are well-defined.
where the polynomial's degree (in x) in the term with denominator nk is 2k.
The definition depends on the unique positive real number
The following arguments demonstrate the equivalence of the above characterizations for the exponential function.
The following argument is adapted from Rudin, theorem 3.31, p. 63–65.
This equivalence can be extended to the negative real numbers by noting
Here, the natural logarithm function is defined in terms of a definite integral as above.
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.)
denote the solution to the initial value problem
Applying the simplest form of Euler's method with increment
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:
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.
is the unique solution of the initial value problem
Indeed, one gets the initial condition f(0) = 1 by dividing both sides of the equation
Assum characterization 5, the multiplicative property together with the initial condition
be a Lebesgue-integrable non-zero function satisfying the mulitiplicative property
Following Hewitt and Stromberg, exercise 18.46, we will prove that Lebesgue-integrability implies continuity.