Exponential function

In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted ⁠\(e^x\)⁠ or ⁠\(\exp x\)⁠; the latter is preferred when the argument ⁠\(x\)⁠ is a complicated expression. It is called exponential because its argument can be seen as an exponent to which a constant number e ≈ 2.718, the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.

The exponential function converts sums to products: ⁠\(\exp(x + y) = \exp x \cdot \exp y\)⁠. Its inverse function, the natural logarithm, ⁠\(\ln\)⁠ or ⁠\(\log\)⁠, converts products to sums: ⁠\(\ln(x\cdot y) = \ln x + \ln y\)⁠.

The exponential function is occasionally called the natural exponential function, matching the name natural logarithm, for distinguishing it from some other functions that are also commonly called exponential functions. These functions include the functions of the form ⁠\(f(x) = b^x\)⁠, which is exponentiation with a fixed base ⁠\(b\)⁠. More generally, and especially in applications, functions of the general form ⁠\(f(x) = ab^x\)⁠ are also called exponential functions. They grow or decay exponentially in that the rate that ⁠\(f(x)\)⁠ changes when ⁠\(x\)⁠ is increased is proportional to the current value of ⁠\(f(x)\)⁠.

The exponential function can be generalized to accept complex numbers as arguments. This reveals relations between multiplication of complex numbers, rotations in the complex plane, and trigonometry. Euler's formula ⁠\(e^{i\theta} = \cos\theta + i\sin\theta\)⁠ expresses and summarizes these relations.

The exponential function can be even further generalized to accept other types of arguments, such as matrices and elements of Lie algebras.

The graph of \(y=e^x\) is upward-sloping, and increases faster than every power of ⁠\(x\)⁠. The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation \(\tfrac{d}{dx}e^x = e^x\) means that the slope of the tangent to the graph at each point is equal to its height (its y-coordinate) at that point.

The exponential function is the unique differentiable function that equals its derivative, and takes the value 1 for the value 0 of its variable.

This definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.

The exponential function is the inverse function of the natural logarithm. That is, \[\begin{align} \ln (\exp x)&=x\\ \exp(\ln y)&=y \end{align}\] for every real number \(x\) and every positive real number \(y.\)

The exponential function is the sum of the power series \[\begin{align}\exp(x) &= 1+x+\frac{x^2}{2!}+ \frac{x^3}{3!}+\cdots\\ &=\sum_{n=0}^\infty \frac{x^n}{n!},\end{align}\] where \(n!\) is the factorial of n (the product of the n first positive integers). This series is absolutely convergent for every \(x\), by the ratio test. This shows that the exponential function is defined for every ⁠\(x\)⁠, and is everywhere the sum of its Maclaurin series.

The exponential satisfies the functional equation \[\exp(x+y)= \exp(x)\cdot \exp(y)\] and maps the additive identity 0 to the multiplicative identity 1. The same equation is satisfied by other continuous functions \(f(x)=b^x\) that exponentiate their argument with an arbitrary base \(b\). Among these functions, the exponential function is characterized by the property that its derivative at 0 is 1.

Reciprocal: The functional equation implies ⁠\(e^x e^{-x}=1\)⁠. Therefore ⁠\(e^x \ne 0\)⁠ for every ⁠\(x\)⁠ and \[\frac 1{e^x}=e^{-x}.\]

Positiveness: ⁠\(e^x>0\)⁠ for every real number ⁠\(x\)⁠. This results from the intermediate value theorem, since ⁠\(e^0=1\)⁠ and, if one would have ⁠\(e^x<0\)⁠ for some ⁠\(x\)⁠, there would be an ⁠\(y\)⁠ such that ⁠\(e^y=0\)⁠ between ⁠\(0\)⁠ and ⁠\(x\)⁠. Since the exponential function equals its derivative, this implies that the exponential function is monotonically increasing.

Extension of exponentiation to positive real bases: Let b be a positive real number. The exponential function and the natural logarithm being the inverse each of the other, one has \(b=\exp(\ln b).\) If n is an integer, the functional equation of the logarithm implies \[b^n=\exp(\ln b^n)= \exp(n\ln b).\] Since the right-most expression is defined if n is any real number, this allows defining ⁠\(b^x\)⁠ for every positive real number b and every real number x: \[b^x=\exp(x\ln b).\] In particular, if b is the Euler's number \(e=\exp(1),\) one has \(\ln e=1\) (inverse function) and thus \[e^x=\exp(x).\] This shows the equivalence of the two notations for the exponential function.

A function is commonly called an exponential function, with an indefinite article, if it has the form ⁠\(x \mapsto b^x\)⁠, that is, if it is obtained from exponentiation by fixing the base and letting the exponent vary.

More generally and especially in applied contexts, the term exponential function is commonly used for functions of the form ⁠\(f(x) = ab^x\)⁠. This may be motivated by the fact that, if the values of the function represent quantities, a change of measurement unit changes the value of ⁠\(a\)⁠, and so, it is nonsensical to impose ⁠\(a=1\)⁠.

These most general exponential functions are the differentiable functions that satisfy the following equivalent characterizations.

• ⁠\(f(x) = ab^x\)⁠ for every ⁠\(x\)⁠ and some constants ⁠\(a\)⁠ and ⁠\(b>0\)⁠.
• ⁠\(f(x)=ae^{kx}\)⁠ for every ⁠\(x\)⁠ and some constants ⁠\(a\)⁠ and ⁠\(k\)⁠.
• The value of \(f'(x)/f(x)\) is independent of \(x\).
• For every \(d,\) the value of \(f(x+d)/f(x)\) is independent of \(x;\) that is, \[\frac{f(x+d)}{f(x)}= \frac{f(y+d)}{f(y)}\] for every x, y.

The base of an exponential function is the base of the exponentiation that appears in it when written as ⁠\(x\to ab^x\)⁠, namely ⁠\(b\)⁠. The base is ⁠\(e^k\)⁠ in the second characterization, \(\exp \frac{f'(x)}{f(x)}\) in the third one, and \(\left(\frac{f(x+d)}{f(x)}\right)^{1/d}\) in the last one.

The last characterization is important in empirical sciences, as allowing a direct experimental test whether a function is an exponential function.

Exponential growth or exponential decay, where the variable change is proportional to the variable value, are thus modeled with exponential functions. Examples are unlimited population growth leading to Malthusian catastrophe, continuously compounded interest, and radioactive decay.

If the modeling function has the form ⁠\(x\mapsto ae^{kx},\)⁠ or, equivalently, is a solution of the differential equation ⁠\(y'=ky\)⁠, the constant ⁠\(k\)⁠ is called, depending on the context, the decay constant, disintegration constant, rate constant, or transformation constant.

For proving the equivalence of the above properties, one can proceed as follows.

The two first characterizations are equivalent, since, if ⁠\(b=e^k\)⁠ and ⁠\(k=\ln b\)⁠, one has \[e^{kx}= (e^k)^x= b^x.\] The basic properties of the exponential function (derivative and functional equation) implies immediately the third and the last condition.

Suppose that the third condition is verified, and let ⁠\(k\)⁠ be the constant value of \(f'(x)/f(x).\) Since \(\frac {\partial e^{kx}}{\partial x}=ke^{kx},\) the quotient rule for derivation implies that \[\frac \partial{\partial x}\,\frac{f(x)}{e^{kx}}=0,\] and thus that there is a constant ⁠\(a\)⁠ such that \(f(x)=ae^{kx}.\)

If the last condition is verified, let \(\varphi(d)=f(x+d)/f(x),\) which is independent of ⁠\(x\)⁠. Using ⁠\(\varphi (0)=1\)⁠, one gets \[\frac{f(x+d)-f(x)}{d} = f(x)\,\frac{\varphi(d)-\varphi(0)}{d}.\] Taking the limit when ⁠\(d\)⁠ tends to zero, one gets that the third condition is verified with ⁠\(k=\varphi'(0)\)⁠. It follows therefore that ⁠\(f(x)= ae^{kx}\)⁠ for some ⁠\(a,\)⁠ and ⁠\(\varphi(d)= e^{kd}.\)⁠ As a byproduct, one gets that \[\left(\frac{f(x+d)}{f(x)}\right)^{1/d}=e^k\] is independent of both ⁠\(x\)⁠ and ⁠\(d\)⁠.

The earliest occurrence of the exponential function was in Jacob Bernoulli's study of compound interests in 1683. This is this study that led Bernoulli to consider the number \[\lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^{n}\] now known as Euler's number and denoted ⁠\(e\)⁠.

The exponential function is involved as follows in the computation of continuously compounded interests.

If a principal amount of 1 earns interest at an annual rate of x compounded monthly, then the interest earned each month is ⁠x/12⁠ times the current value, so each month the total value is multiplied by (1 + ⁠x/12⁠), and the value at the end of the year is (1 + ⁠x/12⁠). If instead interest is compounded daily, this becomes (1 + ⁠x/365⁠). Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, \[\exp x = \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^{n}\] first given by Leonhard Euler.

Exponential functions occur very often in solutions of differential equations.

The exponential functions can be defined as solutions of differential equations. Indeed, the exponential function is a solution of the simplest possible differential equation, namely ⁠\(y'=y\)⁠. Every other exponential function, of the form ⁠\(y=ab^x\)⁠, is a solution of the differential equation ⁠\(y'=ky\)⁠, and every solution of this differential equation has this form.

The solutions of an equation of the form \[y'+ky=f(x)\] involve exponential functions in a more sophisticated way, since they have the form \[y=ce^{-kx}+e^{-kx}\int f(x)e^{kx}dx,\] where ⁠\(c\)⁠ is an arbitrary constant and the integral denotes any antiderivative of its argument.

More generally, the solutions of every linear differential equation with constant coefficients can be expressed in terms of exponential functions and, when they are not homogeneous, antiderivatives. This holds true also for systems of linear differential equations with constant coefficients.

The exponential function can be naturally extended to a complex function, which is a function with the complex numbers as domain and codomain, such that its restriction to the reals is the above-defined exponential function, called real exponential function in what follows. This function is also called the exponential function, and also denoted ⁠\(e^z\)⁠ or ⁠\(\exp(z)\)⁠. For distinguishing the complex case from the real one, the extended function is also called complex exponential function or simply complex exponential.

Most of the definitions of the exponential function can be used verbatim for definiting the complex exponential function, and the proof of their equivalence is the same as in the real case.

The complex exponential function can be defined in several equivalent ways that are the same as in the real case.

The complex exponential is the unique complex function that equals its complex derivative and takes the value ⁠\(1\)⁠ for the argument ⁠\(0\)⁠: \[\frac{de^z}{dz}=e^z\quad\text{and}\quad e^0=1.\]

The complex exponential function is the sum of the series \[e^z = \sum_{k = 0}^\infty\frac{z^k}{k!}.\] This series is absolutely convergent for every complex number ⁠\(z\)⁠. So, the complex exponential is an entire function.

The complex exponential function is the limit \[e^z = \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n\]

As with the real exponential function (see § Functional equation above), the complex exponential satisfies the functional equation \[\exp(z+w)= \exp(z)\cdot \exp(w).\] Among complex functions, it is the unique solution which is holomorphic at the point ⁠\(z = 0\)⁠ and takes the derivative ⁠\(1\)⁠ there.

Condensed: the full section is in Wikipedia.

Complex exponential and trigonometric functions are strongly related by Euler's formula: \[e^{it} =\cos(t)+i\sin(t).\]

This formula provides the decomposition of complex exponentials into real and imaginary parts: \[e^{x+iy} = e^{x}e^{iy} = e^x\,\cos y + i e^x\,\sin y.\]

The trigonometric functions can be expressed in terms of complex exponentials: \[\begin{align} \cos x &= \frac{e^{ix}+e^{-ix}}2\\ \sin x &= \frac{e^{ix}-e^{-ix}}{2i}\\ \tan x &= i\,\frac{1-e^{2ix}}{1+e^{2ix}} \end{align}\]

In these formulas, ⁠\(x, y, t\)⁠ are commonly interpreted as real variables, but the formulas remain valid if the variables are interpreted as complex variables. These formulas may be used to define trigonometric functions of a complex variable.