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".[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 edit
The six most common definitions of the exponential function for real values are as follows.
- Product limit. Define by the limit:
- Power series. Define ex as the value of the infinite series
- Inverse of logarithm integral. Define to be the unique number y > 0 such that
- Differential equation. Define to be the unique solution to the differential equation with initial value:
- Functional equation. The exponential function is the unique function f with 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.
- f is Lebesgue-measurable (Hewitt and Stromberg, 1965, exercise 18.46).
- f is continuous at any one point (Rudin, 1976, chapter 8, exercise 6).
- f is increasing.
- 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 edit
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 function
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 edit
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 edit
The error of the product limit expression is described by:
Characterization 2 edit
Since
Characterization 3 edit
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
Characterization 6 edit
The defnition depends on the unique positive real number satisfying:
Equivalence of the characterizations edit
The following arguments demonstrate the equivalence of the above characterizations for the exponential function.
Characterization 1 ⇔ characterization 2 edit
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,
For the other inequality, by the above expression for tn, if 2 ≤ m ≤ n, we have:
Fix m, and let n approach infinity. Then
This equivalence can be extended to the negative real numbers by noting and taking the limit as n goes to infinity.
Characterization 1 ⇔ characterization 3 edit
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(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 edit
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 1 ⇔ characterization 5 edit
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
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 ⇔ characterization 4 edit
Let n be a non-negative integer. In the sense of definition 4 and by induction, .
Therefore
Using Taylor series,
In the sense of definition 2,
Besides, This shows that definition 2 implies definition 4.
Characterization 2 ⇒ characterization 5 edit
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 edit
Characterisation 3 involves defining the natural logarithm before the exponential function is defined. First,
Characterization 5 ⇒ characterization 4 edit
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
Characterization 5 ⇒ characterization 4 edit
In the sense of definition 5, the multiplicative property together with the initial condition imply that:
Characterization 5 ⇔ characterization 6 edit
The multiplicative property of definition 5 implies that , and that according to the multiplication/division and root definition of exponentiation for rational in definition 6, where . Then the condition means that . Also any of the conditions of definition 5 imply that is continuous at all real . The converse is similar.
References edit
- ^ Walter Rudin (1987). Real and complex analysis (3rd ed.). New York: McGraw-Hill. p. 1. ISBN 978-0-07-054234-1.
- ^ Eli Maor. e: the Story of a Number. p. 156.
- ^ "Herman Yeung - Calculus - First Principle find d/Dx(e^x) 基本原理求 d/Dx(e^x)". YouTube.
- Walter Rudin, Principles of Mathematical Analysis, 3rd edition (McGraw–Hill, 1976), chapter 8.
- Edwin Hewitt and Karl Stromberg, Real and Abstract Analysis (Springer, 1965).