Characterizations of the exponential function


In mathematics, the exponential function can be characterized in many ways. The following characterizations 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, it will be demonstrated that the three most common definitions given for the mathematical constant e are equivalent to each other.

Characterizations

The six most common definitions of the exponential function for real are:

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.,, and all make sense for arbitrary Banach algebras. presents a problem for complex numbers, because there are non-equivalent paths along which one could integrate, and is not sufficient. For example, the function f defined as
satisfies the conditions in without being the exponential function of x + iy. To make 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
In particular, the alternate condition in that is sufficient since it implicitly stipulates that f be conformal.

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, it must be demonstrated that such a limit always exists.

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 is well-defined. It must be shown that the function from to defined by
is a bijection. As is positive for positive t, this function is monotone increasing, hence one-to-one. If the two integrals
hold, then it is clearly onto as well. Indeed, these integrals do hold; they follow from the integral test and the divergence of the harmonic series.

Equivalence of the characterizations

The following proof demonstrates the equivalence of the first 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. Arguments linking the other characterizations are also given.

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,
so that
where ex is in the sense of definition 2. Here, limsups must be used, because it is not known if tn converges. For the other direction, by the above expression of tn, if 2 ≤ m ≤ n,
Fix m, and let n approach infinity. Then
. Now, taking the above inequality, letting m approach infinity, and putting it together with the other inequality, this becomes
so that
This equivalence can be extended 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 the term with denominator nk is 2k.

Equivalence of characterizations 1 and 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,
Besides,
Now, let x be any fixed real number, and let
Ln = x, which implies that y = ex, where ex is in the sense of definition 3. We have
Here, the continuity of ln 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.

Equivalence of characterizations 2 and 4

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.

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 implies continuity for a non-zero function satisfying, and then one proves that continuity implies for some k, and finally implies k=1.
First, a few elementary properties from satisfying are proven, and the assumption that is not identically zero:
The second and third properties mean that it is sufficient to prove for positive x.
If is a Lebesgue-integrable function, then
It then follows that
Since is nonzero, some y can be chosen 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, can be proven, 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. If restricted 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 of the rationals is the reals. If then k = 1. This is equivalent to characterization 1, depending on which equivalent definition of e one uses.

Characterization 2 implies characterization 6

In the sense of definition 2,

Characterization 5 implies characterization 4

Characterization 6 implies characterization 4

In the sense of definition 6,
By the way, therefore definition 6 implies definition 4.