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.

Characterizations

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

1. Define e^x by the limit
   $${\displaystyle e^{x}=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.}$$

    
2. Define e^x as the value of the infinite series
   $${\displaystyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots }$$

    
   (Here n! denotes the factorial of n. One proof that e is irrational uses a special case of this formula.)
3. Define e^x to be the unique number y > 0 such that
   $${\displaystyle \int _{1}^{y}{\frac {dt}{t}}=x.}$$

    
   This is as the inverse of the natural logarithm function, which is defined by this integral.
4. Define e^x to be the unique solution to the initial value problem
   $${\displaystyle y'=y,\quad y(0)=1.}$$

    
   (Here, y′ denotes the derivative of y.)
5. The exponential function e^x 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 e^x on rational numbers must equal e^x.)
   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
   $${\displaystyle \lim _{h\to 0}{\frac {e^{h}-1}{h}}=1.}$$

    
   This limit can be shown to exist. Then define e^x to be the exponential function with this base. This definition is particularly suited to computing the derivative of the exponential function.

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. (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

$${\displaystyle f(x+iy)=e^{x}(\cos(2y)+i\sin(2y))=e^{x+2iy}}$$

 

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

$${\displaystyle f(i)=\cos(1)+i\sin(1).}$$

 

In particular, the alternate condition in (5) that ${\displaystyle f'(0)=1}$  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.

Characterization 2

Since

$${\displaystyle \lim _{n\to \infty }\left|{\frac {x^{n+1}/(n+1)!}{x^{n}/n!}}\right|=\lim _{n\to \infty }\left|{\frac {x}{n+1}}\right|=0<1.}$$

 

it follows from the ratio test that ${\textstyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$  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 ${\displaystyle \mathbb {R} ^{+}}$  to ${\displaystyle \mathbb {R} }$  defined by

$${\displaystyle x\mapsto \int _{1}^{x}{\frac {dt}{t}}}$$

 

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

$${\displaystyle {\begin{aligned}\int _{1}^{\infty }{\frac {dt}{t}}&=\infty \\[8pt]\int _{1}^{0}{\frac {dt}{t}}&=-\infty \end{aligned}}}$$

 

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.

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.

Characterization 1 ⇔ characterization 2

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

Let ${\displaystyle x\geq 0}$  be a fixed non-negative real number. Define

$${\displaystyle s_{n}=\sum _{k=0}^{n}{\frac {x^{k}}{k!}},\ t_{n}=\left(1+{\frac {x}{n}}\right)^{n}.}$$

 

By the binomial theorem,

$${\displaystyle {\begin{aligned}t_{n}&=\sum _{k=0}^{n}{n \choose k}{\frac {x^{k}}{n^{k}}}=1+x+\sum _{k=2}^{n}{\frac {n(n-1)(n-2)\cdots (n-(k-1))x^{k}}{k!\,n^{k}}}\\[8pt]&=1+x+{\frac {x^{2}}{2!}}\left(1-{\frac {1}{n}}\right)+{\frac {x^{3}}{3!}}\left(1-{\frac {1}{n}}\right)\left(1-{\frac {2}{n}}\right)+\cdots \\[8pt]&{}\qquad \cdots +{\frac {x^{n}}{n!}}\left(1-{\frac {1}{n}}\right)\cdots \left(1-{\frac {n-1}{n}}\right)\leq s_{n}\end{aligned}}}$$

 

(using x ≥ 0 to obtain the final inequality) so that

$${\displaystyle \limsup _{n\to \infty }t_{n}\leq \limsup _{n\to \infty }s_{n}=e^{x}}$$

 

where e^x is in the sense of definition 2. Here, limsups must be used, because it is not known if t_n converges. For the other direction, by the above expression of t_n, if 2 ≤ m ≤ n,

$${\displaystyle 1+x+{\frac {x^{2}}{2!}}\left(1-{\frac {1}{n}}\right)+\cdots +{\frac {x^{m}}{m!}}\left(1-{\frac {1}{n}}\right)\left(1-{\frac {2}{n}}\right)\cdots \left(1-{\frac {m-1}{n}}\right)\leq t_{n}.}$$

 

Fix m, and let n approach infinity. Then

$${\displaystyle s_{m}=1+x+{\frac {x^{2}}{2!}}+\cdots +{\frac {x^{m}}{m!}}\leq \liminf _{n\to \infty }t_{n}}$$

 

(again, liminf's must be used because it is not known if t_n converges). Now, taking the above inequality, letting m approach infinity, and putting it together with the other inequality, this becomes

$${\displaystyle \limsup _{n\to \infty }t_{n}\leq e^{x}\leq \liminf _{n\to \infty }t_{n}}$$

 

so that

$${\displaystyle \lim _{n\to \infty }t_{n}=e^{x}.}$$

 

This equivalence can be extended to the negative real numbers by noting ${\textstyle \left(1-{\frac {r}{n}}\right)^{n}\left(1+{\frac {r}{n}}\right)^{n}=\left(1-{\frac {r^{2}}{n^{2}}}\right)^{n}}$  and taking the limit as n goes to infinity.

The error term of this limit-expression is described by

$${\displaystyle \left(1+{\frac {x}{n}}\right)^{n}=e^{x}\left(1-{\frac {x^{2}}{2n}}+{\frac {x^{3}(8+3x)}{24n^{2}}}+\cdots \right),}$$

 

where the polynomial's degree (in x) in the term with denominator n^k is 2k.

Characterization 1 ⇔ characterization 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,

$${\displaystyle {\frac {d}{dx}}\ln x={\frac {d}{dx}}\int _{1}^{x}{\frac {1}{t}}\,dt={\frac {1}{x}}.}$$

 

Besides, ${\textstyle \ln 1=\int _{1}^{1}{\frac {dt}{t}}=0}$ 

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

$${\displaystyle y=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.}$$

 

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

$${\displaystyle \ln y=\ln \lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}=\lim _{n\to \infty }\ln \left(1+{\frac {x}{n}}\right)^{n}.}$$

 

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

$${\displaystyle \ln y=\lim _{n\to \infty }n\ln \left(1+{\frac {x}{n}}\right)=\lim _{n\to \infty }{\frac {x\ln \left(1+(x/n)\right)}{(x/n)}}.}$$

 

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 a^b can be defined for real b as e^{b lna}.)

$${\displaystyle =x\cdot \lim _{h\to 0}{\frac {\ln \left(1+h\right)}{h}}\quad {\text{ where }}h={\frac {x}{n}}}$$

 

$${\displaystyle =x\cdot \lim _{h\to 0}{\frac {\ln \left(1+h\right)-\ln 1}{h}}}$$

 

$${\displaystyle =x\cdot {\frac {d}{dt}}\ln t{\Bigg |}_{t=1}}$$

 

$${\displaystyle \!\,=x.}$$

 

Characterization 1 ⇔ characterization 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 ${\displaystyle f(x)}$  satisfying ${\displaystyle f(x+y)=f(x)f(y)}$ , and then one proves that continuity implies ${\displaystyle f(x)=e^{kx}}$  for some k, and finally ${\displaystyle f(1)=e}$  implies k = 1.

First, a few elementary properties from ${\displaystyle f(x)}$  satisfying ${\displaystyle f(x+y)=f(x)f(y)}$  are proven, and the assumption that ${\displaystyle f(x)}$  is not identically zero:

• If ${\displaystyle f(x)}$  is nonzero anywhere (say at x=y), then it is non-zero everywhere. Proof: ${\displaystyle f(y)=f(x)f(y-x)\neq 0}$  implies ${\displaystyle f(x)\neq 0}$ .
• ${\displaystyle f(0)=1}$ . Proof: ${\displaystyle f(x)=f(x+0)=f(x)f(0)}$  and ${\displaystyle f(x)}$  is non-zero.
• ${\displaystyle f(-x)=1/f(x)}$ . Proof: ${\displaystyle 1=f(0)=f(x-x)=f(x)f(-x)}$ .
• If ${\displaystyle f(x)}$  is continuous anywhere (say at x = y), then it is continuous everywhere. Proof: ${\displaystyle f(x+\delta )-f(x)=f(x-y)[f(y+\delta )-f(y)]\to 0}$  as ${\displaystyle \delta \to 0}$  by continuity at y.

The second and third properties mean that it is sufficient to prove ${\displaystyle f(x)=e^{x}}$  for positive x.

If ${\displaystyle f(x)}$  is a Lebesgue-integrable function, then

$${\displaystyle g(x)=\int _{0}^{x}f(x')\,dx'.}$$

 

It then follows that

$${\displaystyle g(x+y)-g(x)=\int _{x}^{x+y}f(x')\,dx'=\int _{0}^{y}f(x+x')\,dx'=f(x)g(y).}$$

 

Since ${\displaystyle f(x)}$  is nonzero, some y can be chosen such that ${\displaystyle g(y)\neq 0}$  and solve for ${\displaystyle f(x)}$  in the above expression. Therefore:

$${\displaystyle {\begin{aligned}f(x+\delta )-f(x)&={\frac {[g(x+\delta +y)-g(x+\delta )]-[g(x+y)-g(x)]}{g(y)}}\\&={\frac {[g(x+y+\delta )-g(x+y)]-[g(x+\delta )-g(x)]}{g(y)}}\\&={\frac {f(x+y)g(\delta )-f(x)g(\delta )}{g(y)}}=g(\delta ){\frac {f(x+y)-f(x)}{g(y)}}.\end{aligned}}}$$

 

The final expression must go to zero as ${\displaystyle \delta \to 0}$  since ${\displaystyle g(0)=0}$  and ${\displaystyle g(x)}$  is continuous. It follows that ${\displaystyle f(x)}$  is continuous.

Now, ${\displaystyle f(q)=e^{kq}}$  can be proven, for some k, for all positive rational numbers q. Let q=n/m for positive integers n and m. Then

$${\displaystyle f\left({\frac {n}{m}}\right)=f\left({\frac {1}{m}}+\cdots +{\frac {1}{m}}\right)=f\left({\frac {1}{m}}\right)^{n}}$$

 

by elementary induction on n. Therefore, ${\displaystyle f(1/m)^{m}=f(1)}$  and thus

$${\displaystyle f\left({\frac {n}{m}}\right)=f(1)^{n/m}=e^{k(n/m)}.}$$

 

for ${\displaystyle k=\ln[f(1)]}$ . If restricted to real-valued ${\displaystyle f(x)}$ , then ${\displaystyle f(x)=f(x/2)^{2}}$  is everywhere positive and so k is real.

Finally, by continuity, since ${\displaystyle f(x)=e^{kx}}$  for all rational x, it must be true for all real x since the closure of the rationals is the reals (that is, any real x can be written as the limit of a sequence of rationals). If ${\displaystyle f(1)=e}$  then k = 1. This is equivalent to characterization 1 (or 2, or 3), depending on which equivalent definition of e one uses.

Characterization 2 ⇔ characterization 4

Let n be a non-negative integer. In the sense of definition 4 and by induction, ${\displaystyle {\frac {d^{n}y}{dx^{n}}}=y}$ .

Therefore ${\displaystyle {\frac {d^{n}y}{dx^{n}}}{\Bigg |}_{x=0}=y(0)=1.}$ 

Using Taylor series,

$${\displaystyle y=\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}\,x^{n}=\sum _{n=0}^{\infty }{\frac {1}{n!}}\,x^{n}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}$$

 

This shows that definition 4 implies definition 2.

In the sense of definition 2,

$${\displaystyle {\begin{aligned}{\frac {d}{dx}}e^{x}&={\frac {d}{dx}}\left(1+\sum _{n=1}^{\infty }{\frac {x^{n}}{n!}}\right)=\sum _{n=1}^{\infty }{\frac {nx^{n-1}}{n!}}=\sum _{n=1}^{\infty }{\frac {x^{n-1}}{(n-1)!}}\\[6pt]&=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}},{\text{ where }}k=n-1\\[6pt]&=e^{x}\end{aligned}}}$$

 

Besides, ${\textstyle e^{0}=1+0+{\frac {0^{2}}{2!}}+{\frac {0^{3}}{3!}}+\cdots =1.}$  This shows that definition 2 implies definition 4.

Characterization 2 ⇒ characterization 6

In the sense of definition 2,^[1]

$${\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {e^{h}-1}{h}}&=\lim _{h\to 0}{\frac {1}{h}}\left(\left(1+h+{\frac {h^{2}}{2!}}+{\frac {h^{3}}{3!}}+{\frac {h^{4}}{4!}}+\cdots \right)-1\right)\\&=\lim _{h\to 0}\left(1+{\frac {h}{2!}}+{\frac {h^{2}}{3!}}+{\frac {h^{3}}{4!}}+\cdots \right)\\&=1\end{aligned}}}$$

 

Characterization 3 ⇔ characterization 4

Characterisation 3 involves defining the natural logarithm before the exponential function is defined. First,

$${\displaystyle \log x:=\int _{1}^{x}{\frac {dt}{t}}}$$

 

This means that the natural logarithm of ${\displaystyle x}$  equals the (signed) area under the graph of ${\displaystyle 1/t}$  between ${\displaystyle t=1}$  and ${\displaystyle t=x}$ . If ${\displaystyle x<1}$ , then this area is taken to be negative. Then, ${\displaystyle \exp }$  is defined as the inverse of ${\displaystyle \log }$ , meaning that

$${\displaystyle \exp(\log(x))=x{\text{ and }}\log(\exp(x))=x}$$

 

by the definition of an inverse function. If ${\displaystyle a}$  is a positive real number then ${\displaystyle a^{x}}$  is defined as ${\displaystyle \exp(x\log(a))}$ . Finally, ${\displaystyle e}$  is defined as the number ${\displaystyle a}$  such that ${\displaystyle \log(a)=1}$ . It can then be shown that ${\displaystyle e^{x}=\exp(x)}$ :

$${\displaystyle e^{x}=\exp(x\log(e))=\exp(x)}$$

 

By the fundamental theorem of calculus, the derivative of ${\textstyle \log x={\frac {1}{x}}}$ . We are now in a position to prove that ${\textstyle {\frac {d}{dx}}e^{x}=e^{x}}$ , satisfying the first part of the initial value problem given in characterisation 4:

$${\displaystyle {\begin{aligned}{\text{Let }}y&=e^{x}=\exp(x)\\\log(y)&=\log(\exp(x))=x\\{\frac {1}{y}}{\frac {dy}{dx}}&=1\\{\frac {dy}{dx}}&=y=e^{x}\end{aligned}}}$$

 

Then, we merely have to note that ${\displaystyle e^{0}=\exp(0)=1}$ , and we are done. Of course, it is much easier to show that characterisation 4 implies characterisation 3. If ${\displaystyle e^{x}}$  is the unique function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$  satisfying ${\displaystyle f'(x)=e^{x}}$ , and ${\displaystyle f(0)=1}$ , then ${\displaystyle \log }$  can be defined as its inverse. The derivative of ${\displaystyle \log }$  can be found in the following way:

$${\displaystyle y=\log x\implies x=e^{y}}$$

 

If we differentiate both sides with respect to ${\displaystyle y}$ , we get

$${\displaystyle {\begin{aligned}{\frac {dx}{dy}}&=e^{y}\\{\frac {dy}{dx}}&={\frac {1}{e^{y}}}={\frac {1}{x}}\end{aligned}}}$$

 

Therefore,

$${\displaystyle \int _{1}^{x}{\frac {1}{t}}dt=\left[\log t\right]_{1}^{x}=\log x-\log 1=\log x-0=\log x}$$

 

Characterization 5 ⇒ 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

$${\displaystyle f(0)=f(0+0)=f(0)f(0)}$$

 

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:

$${\displaystyle {\begin{array}{rcccccc}f'(x)&=&\lim \limits _{h\to 0}{\frac {f(x+h)-f(x)}{h}}&=&\lim \limits _{h\to 0}{\frac {f(x)f(h)-f(x)}{h}}&=&\lim \limits _{h\to 0}f(x){\frac {f(h)-1}{h}}\\[1em]&=&f(x)\lim \limits _{h\to 0}{\frac {f(h)-1}{h}}&=&f(x)\lim \limits _{h\to 0}{\frac {f(0+h)-f(0)}{h}}&=&f(x)f'(0)=f(x).\end{array}}}$$

 

Characterization 6 ⇒ characterization 4

In the sense of definition 6,

$${\displaystyle {\frac {d}{dx}}e^{x}=\lim _{h\to 0}{\frac {e^{x+h}-e^{x}}{h}}=e^{x}\cdot \lim _{h\to 0}{\frac {e^{h}-1}{h}}=e^{x}.}$$

 

By the way ${\displaystyle e^{0}=1}$ , therefore definition 6 implies definition 4.

References

1. "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).