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, we will see that the three most common definitions given for the mathematical constant e are also equivalent to each other. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The exponential function is one of the most important functions in mathematics. ... In the jargon of mathematics, the statement that Property P characterizes object X means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as Property Q characterises Y up to isomorphism. The first... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ...

Characterizations

The five most common definitions of the exponential function exp(x) = e^x are:

1. Define e^x by the limit

2. Define e^x as the sum of the infinite series

(Here, n! stands for the factorial of n. The proof that e is irrational uses this representation.)

3. Define e^x to be the unique number y > 0 such that

4. Define e^x to be the unique solution to the initial value problem

5. The exponential function f(x) = e^x is the unique Lebesgue-measurable function with f(1) = e that satisfies:

(Hewitt and Stromberg, 1965, exercise 18.46). Alternatively, it is the unique anywhere-continuous function with these properties (Rudin, 1976, chapter 8 exercise 6). As another alternative (as long as the domain is assumed to contain only real numbers), it is the only monotonic function satisfying those identities. (As a counter-example, if one does not assume continuity or measurability, it is possible to prove the existence of an everywhere-discontinuous, non-measurable function with this property by using a Hamel basis for the real numbers over the rationals, as described in Hewitt and Stromberg.)

These definitions are not limited to the exponential of real numbers, and in several cases can be extended to any Banach algebra. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as... In mathematics, a series is a sum of a sequence of terms. ... For factorial rings in mathematics, see unique factorisation domain. ... In mathematics, the series expansion of the number e can be used to prove that e is irrational. ... In mathematics, an initial value problem is a statement of a differential equation together with specified value of the unknown function at a given point in the domain of the solution. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ...

Real versus complex

The characterizations above are equivalent if the domain is taken to be the set of all real numbers. However, if one takes the domain to be the complex numbers, then conditions (1), (2), and (4) are sufficient, but (3) is problematic (along which path does one integrate?) and (5) is not sufficient. This last statement means that some functions satisfy the functional equation In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ...

for all complex numbers z and w, and the initial condition

and are continuous, but nonetheless do not coincide with the natural exponential function. For example, for x and y real, let

That this function is not differentiable in the complex sense follows from the fact that it fails to satisfy the Cauchy-Riemann equations. In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic. ...

To make (5) sufficient, one may either stipulate that there exists a point at which f is a conformal map or else stipulate that In mathematics, a conformal map is a function which preserves angles. ...

Why each characterization makes sense

Each characterization requires some justification to show that it makes sense. For instance, when the value of the function is defined by a sequence or series, the convergence of this sequence or series needs to be established.

Characterization 1

It can be shown that the sequence

is an increasing sequence which is bounded above. Since every bounded, increasing sequence of real numbers converges to a unique real number, this characterization makes sense.

Characterization 2

To show the infinite series converges at x = 1, it is enough to compare with a geometric series: In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

To show that the series converges for all x, we use the ratio test, which shows that the series has an infinite radius of convergence, since In mathematics, the ratio test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. ...

Characterization 3

In this case, we define the natural logarithm function ln(x) first, and then define exp(x) as the inverse of the natural logarithm. In other words, for all y > 0, define The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ...

Since 1/t is continuous for all t > 0, this function makes sense, and since 1/t is positive for all t > 0, this function is strictly increasing (hence, injective) for y > 0. (Note that if y < 1, then ln(y) is a negative number.) By the integral test and the divergence of the harmonic series, it follows that ln(y) → ∞ as y → ∞. By a similar argument, a change of variables (t 1/t) shows that ln(y) → −∞ as y → 0. To sum up, ln(y) maps the infinite interval (0, ∞) bijectively onto the real line (−∞, ∞). Hence, for any real number x, there must exist a unique number y > 0 such that ln(y) = x. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... See harmonic series (music) for the (related) musical concept. ... A bijective function. ... In mathematics, the real line is simply the set of real numbers. ...

Equivalence of the characterizations

The following proof demonstrates the equivalence of the 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.

Equivalence of characterizations 1 and 2

The following argument is adapted from a proof in Rudin, theorem 3.31, p. 63-65.

Let x be a fixed real number. Define

By the binomial theorem, In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

so that

where e^x is in the sense of definition 2. Here, we must use limsup's, because we don't yet know that t_n actually converges. Now, for the other direction, note that by the above expression of t_n, if 2 ≤ m ≤ n, we have In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...

Fix m, and let n approach infinity. We get

(again, we must use liminf's because we don't yet know that t_n converges). Now, take the above inequality, let m approach infinity, and put it together with the other inequality. This becomes In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. ...

so that

Equivalence of characterizations 1 and 3

Here, we define the natural logarithm function in terms of a definite integral as above. By the fundamental theorem of calculus, The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ...

Now, let x be any fixed real number, and let

We will show that ln(y) = x, which implies that y = e^x, where e^x is in the sense of definition 3. We have

Here, we have used the continuity of ln(y), which follows from the continuity of 1/t:

Here, we have used the result lnaⁿ = nlna. 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 a^b can be defined for real b as e^{b lna}.)

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 f(x) satisfying f(x + y) = f(x)f(y), and then one proves that continuity implies f(x) = e^kx for some k, and finally f(1) = e implies k=1.

First, we prove a few elementary properties from f(x) satisfying f(x + y) = f(x)f(y) and the assumption that f(x) is not identically zero:

• If f(x) is nonzero anywhere (say at x=y), then it is non-zero everywhere. Proof: implies .
• f(0) = 1. Proof: f(x) = f(x + 0) = f(x)f(0) and f(x) is non-zero.
• f( − x) = 1 / f(x). Proof: 1 = f(0) = f(x − x) = f(x)f( − x).
• If f(x) 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 f(x) = e^x for positive x.

If f(x) is a Lebesgue-integrable function, then we can define The integral of a positive function can be interpreted as the area under a curve. ...

It then follows that

Since f(x) is nonzero, we can choose some y such that and solve for f(x) in the above expression. Therefore:

The final expression must go to zero as since g(0) = 0 and g(x) is continuous. It follows that f(x) is continuous.

Now, we prove that f(q) = e^kq, 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, f(1 / m)^m = f(1) and thus

for . Note that if we are restricting ourselves to real-valued f(x), then f(x) = f(x / 2)² is everywhere positive and so k is real.

Finally, by continuity, since f(x) = e^kx for all rational x, it must be true for all real x since the closure of the rationals is the reals (that is, we can write any real x as the limit of a sequence of rationals). If f(1) = e then k = 1. This is equivalent to characterization 1 (or 2, or 3), depending on which equivalent definition of e one uses. In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ...

References

• Walter Rudin, Principles of Mathematical Analysis, 3rd edition (McGraw-Hill, 1976), chapter 8.
• Edwin Hewitt and Karl Stromberg, Real and Abstract Analysis (Springer, 1965).