Characterizations of the exponential function

Summary

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

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The six most common definitions of the exponential function   for real values   are as follows.

  1. Product limit. Define   by the limit: 
  2. Power series. Define ex as the value of the infinite series   (Here n! denotes the factorial of n. One proof that e is irrational uses a special case of this formula.)
  3. Inverse of logarithm integral. Define   to be the unique number y > 0 such that   That is,   is the inverse of the natural logarithm function  , which is defined by this integral.
  4. Differential equation. Define   to be the unique solution to the differential equation with initial value:  where   denotes the derivative of y.
  5. Functional equation. The exponential function   is the unique function f with the multiplicative property   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.
  6. 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

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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 satisfies the three listed regularity conditions in (5) but is not equal to  . A sufficient condition is that   and that   is a conformal map at some point; or else the two initial values   and   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

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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

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The error of the product limit expression is described by:  where the polynomial's degree (in x) in the term with denominator nk is 2k.

Characterization 2

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Since   it follows from the ratio test that   converges for all x.

Characterization 3

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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. Since 1/t is positive for positive t, this function is strictly increasing, hence injective. If the two integrals   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

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The definition depends on the unique positive real number   satisfying:  This limit can be shown to exist for any  , and it defines a continuous increasing function   with   and  , so the Intermediate value theorem guarantees the existence of such a value  .

Equivalence of the characterizations

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

Characterization 1 ⇔ characterization 2

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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,   (using x ≥ 0 to obtain the final inequality) so that:   One must use lim sup because it is not known if tn converges.

For the other inequality, by the above expression for tn, if 2 ≤ mn, we have:  

Fix m, and let n approach infinity. Then   (again, one must use lim inf because it is not known if tn converges). Now, take the above inequality, let m approach infinity, and put it together with the other inequality to obtain:   so that  

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

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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

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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 2 ⇔ characterization 4

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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.

Characterization 2 ⇒ characterization 5

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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

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Characterisation 3 first defines the natural logarithm: then   as the inverse function with  . Then by the Chain rule:

 

i.e.  . Finally,  , so  . That is,   is the unique solution of the initial value problem  ,   of characterization 4. Conversely, assume   has   and  , and define   as its inverse function with   and  . Then:

 

i.e.  . By the Fundamental theorem of calculus, 

Characterization 5 ⇒ characterization 4

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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 5 ⇒ characterization 4

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Assum characterization 5, the multiplicative property together with the initial condition   imply that:  

Characterization 5 ⇔ characterization 6

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By inductively applying the multiplication rule, we get:   and thus   for  . Then the condition   means that  , so   by definition.

Also, any of the regularity conditions of definition 5 imply that   is continuous at all real   (see below). The converse is similar.

Characterization 5 ⇒ characterization 6

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Let   be a Lebesgue-integrable non-zero function satisfying the mulitiplicative property   with  . Following Hewitt and Stromberg, exercise 18.46, we will prove that Lebesgue-integrability implies continuity. This is sufficient to imply   according to characterization 6, arguing as above.

First, a few elementary properties:

  1. If   is nonzero anywhere (say at  ), then it is non-zero everywhere. Proof:   implies  .
  2.  . Proof:   and   is non-zero.
  3.  . Proof:  .
  4. If   is continuous anywhere (say at  ), then it is continuous everywhere. Proof:   as   by continuity at  .

The second and third properties mean that it is sufficient to prove   for positive x.

Since  is a Lebesgue-integrable function, then we may define  . 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.

References

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  1. ^ Walter Rudin (1987). Real and complex analysis (3rd ed.). New York: McGraw-Hill. p. 1. ISBN 978-0-07-054234-1.
  2. ^ Eli Maor. e: the Story of a Number. p. 156.
  3. ^ "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).