Characterizations of the exponential function
In mathematics, the exponential function can be characterized 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, it will be demonstrated that the three most common definitions given for the mathematical constant e are equivalent to each other.
The six most common definitions of the exponential function exp(x) = ex for real x are:
- 1. Define ex by the limit
- 2. Define ex as the value of the infinite series
- 3. Define ex to be the unique number y > 0 such that
- This is as the inverse of the natural logarithm function, which is defined by this integral.
- 4. Define ex to be the unique solution to the initial value problem
- (Here, y′ denotes the derivative of y.)
- 5. The exponential function ex is the unique function f with f(1) = e and f(x + y) = f(x) f(y) for all x and y that satisfies any one of the following additional conditions:
- f is Lebesgue-measurable (Hewitt and Stromberg, 1965, exercise 18.46).
- f is continuous at at least one point (Rudin, 1976, chapter 8, exercise 6). (As shown below, if f(x + y) = f(x) f(y) for all x and y, and f is continuous at any single point, then f is necessarily continuous everywhere.)
- f is increasing. (An increasing function that agrees with ex on rational numbers must equal ex.)
- For the uniqueness, one must impose some additional condition like those above, since otherwise other functions can be constructed using a basis for the real numbers over the rationals, as described by Hewitt and Stromberg.
- One could also replace f(1) = e and the "additional condition" with the single condition f′(0) = 1.
- 6. Let e be the unique positive real number satisfying
- This limit can be shown to exist. Then define ex to be the exponential function with this base. This definition is particularly suited to computing the derivative of the exponential function.
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. (1), (2), and (4) all make sense for arbitrary Banach algebras. (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 or else stipulate that
In particular, the alternate condition in (5) 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 (i.e. an infinite sequence or series), it must be demonstrated that such a limit always exists.
it follows from the ratio test that converges for all x.
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
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,
(using x ≥ 0 to obtain the final inequality) so that
Fix m, and let n approach infinity. Then
(again, liminf's must be used because it is not known if tn converges). Now, taking the above inequality, letting m approach infinity, and putting it together with the other inequality, this becomes
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 x) in the term with denominator nk is 2k.
Equivalence of characterizations 1 and 3
Now, let x be any fixed real number, and let
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.)
Equivalence of characterizations 3 and 4
Characterisation 3 involves defining the natural logarithm before the exponential function is defined. First,
This means that the natural logarithm of equals the (signed) area under the graph of between and . If , then this area is taken to be negative. Then, is defined as the inverse of , meaning that
by the definition of an inverse function. If is a positive real number then is defined as . Finally, is defined as the number such that . It can then be shown that :
By the fundamental theorem of calculus, the derivative of . We are now in a position to prove that , satisfying the first part of the initial value problem given in characterisation 4:
Then, we merely have to note that , and we are done. Of course, it is much easier to show that characterisation 4 implies characterisation 3. If is the unique function satisfying , and , then can be defined as its inverse. The derivative of can be found in the following way:
If we differentiate both sides with respect to , we get
Equivalence of characterizations 2 and 4
Let n be a non-negative integer. In the sense of definition 4 and by induction, .
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 (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, a few elementary properties from satisfying are proven, 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
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 (that is, any real x can be written 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 one uses.
Characterization 2 implies characterization 6
In the sense of definition 2,
Characterization 5 implies characterization 4
- 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 6 implies characterization 4
In the sense of definition 6, By the way , therefore definition 6 implies definition 4.