Exponential function: Difference between revisions

Latest revision as of 19:52, 14 February 2026

Short description: Mathematical function, denoted exp(x) or e^x

In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. The exponential of a variable $x$ is denoted $\exp x$ or $e^{x}$ , with the two notations used interchangeably. It is called exponential because its argument can be seen as an exponent to which a constant number $e \approx 2.718$ , 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: it maps the additive identity $0$ to the multiplicative identity $1$ , and the exponential of a sum is equal to the product of separate exponentials, $\exp (x + y) = \exp x \cdot \exp y$ . 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 $f (x) = a b^{x}$ 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 $\exp i θ = \cos θ + i \sin θ$ 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.

Graph

The graph of $y = e^{x}$ is upward-sloping, and increases faster than every power of $x$ .^[1] 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 $\frac{d}{d x} e^{x} = e^{x}$ means that the slope of the tangent to the graph at each point is equal to its height (its $y$ -coordinate) at that point.

Definitions and fundamental properties

There are several equivalent definitions of the exponential function, although of very different nature.

Differential equation

One of the simplest definitions is: 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 "conceptual" definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.

Uniqueness: If $f (x)$ and $g (x)$ are two functions satisfying the above definition, then the derivative of $f / g$ is zero everywhere because of the quotient rule. It follows that $f / g$ is constant; this constant is $1$ since $f (0) = g (0) = 1$ .

Existence is proved in each of the two following sections.

Inverse of natural logarithm

The exponential function is the inverse function of the natural logarithm. The inverse function theorem implies that the natural logarithm has an inverse function, that satisfies the above definition. This is a first proof of existence. Therefore, one has

\begin{aligned} \ln (\exp x) & = x \\ \exp (\ln y) & = y \end{aligned}

for every real number $x$ and every positive real number $y .$

Power series

The exponential function is the sum of the power series^[2]^[3] $\begin{aligned} \exp (x) & = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots \\ = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}, \end{aligned}$

where $n!$ is the factorial of $n$ (the product of the $n$ first positive integers). This series is absolutely convergent for every $x$ per the ratio test. So, the derivative of the sum can be computed by term-by-term differentiation, and this shows that the sum of the series satisfies the above definition. This is a second existence proof, and shows, as a byproduct, that the exponential function is defined for every $x$ , and is everywhere the sum of its Maclaurin series.

Functional equation

The exponential satisfies the functional equation: $\exp (x + y) = \exp (x) \cdot \exp (y) .$ This results from the uniqueness and the fact that the function $f (x) = \exp (x + y) / \exp (y)$ satisfies the above definition.

It can be proved that a function that satisfies this functional equation has the form $x \mapsto \exp (c x)$ if it is either continuous or monotonic. It is thus differentiable, and equals the exponential function if its derivative at $0$ is $1$ .

Limit of integer powers

The exponential function is the limit, as the integer $n$ goes to infinity,^[4]^[3] $\exp (x) = \lim_{n \to + \infty} {(1 + \frac{x}{n})}^{n} .$ By continuity of the logarithm, this can be proved by taking logarithms and proving $x = \lim_{n \to \infty} \ln {(1 + \frac{x}{n})}^{n} = \lim_{n \to \infty} n \ln (1 + \frac{x}{n}),$ for example with Taylor's theorem.

Properties

Reciprocal: The functional equation implies $e^{x} e^{- x} = 1$ . Therefore $e^{x} \neq 0$ 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.

General exponential functions

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 $f (x) = a b^{x}$ . 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.

$f (x) = a b^{x}$ for every $x$ and some constants $a$ and $b > 0$ .
$f (x) = a e^{k x}$ 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 $x$ , $y$ .^[5]

The base of an exponential function is the base of the exponentiation that appears in it when written as $x \to a b^{x}$ , namely $b$ .^[6] The base is $e^{k}$ in the second characterization, $\exp \frac{f^{'} (x)}{f (x)}$ in the third one, and ${(\frac{f (x + d)}{f (x)})}^{1 / d}$ in the last one.

In applications

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 $x \mapsto a e^{k x},$ or, equivalently, is a solution of the differential equation $y^{'} = k y$ , the constant $k$ is called, depending on the context, the decay constant, disintegration constant,^[7] rate constant,^[8] or transformation constant.^[9]

Equivalence proof

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 $e^{k x} = (e^{k})^{x} = b^{x} .$ 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 $\frac{\partial e^{k x}}{\partial x} = k e^{k x},$ the quotient rule for derivation implies that $\frac{\partial}{\partial x} \frac{f (x)}{e^{k x}} = 0,$ and thus that there is a constant $a$ such that $f (x) = a e^{k x} .$

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

Compound interest

The earliest occurrence of the exponential function was in Jacob Bernoulli's study of compound interests in 1683.^[10] This is this study that led Bernoulli to consider the number $\lim_{n \to \infty} {(1 + \frac{1}{n})}^{n}$ 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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠x/12⁠$ 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 $(1 + ⁠ x / 12 ⁠) 12$ . If instead interest is compounded daily, this becomes $(1 + ⁠ x / 365 ⁠) 365$ . Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, $\exp x = \lim_{n \to \infty} {(1 + \frac{x}{n})}^{n}$ first given by Leonhard Euler.^[4]

Differential equations

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 $y = a b^{x}$ , is a solution of the differential equation $y^{'} = k y$ , and every solution of this differential equation has this form.

The solutions of an equation of the form $y^{'} + k y = f (x)$ involve exponential functions in a more sophisticated way, since they have the form $y = c e^{- k x} + e^{- k x} \int f (x) e^{k x} d x,$ 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.

Complex exponential

A complex plot of $z \mapsto \exp z$ , with the argument $Arg \exp z$ represented by varying hue. The transition from dark to light colors shows that $| \exp z |$ is increasing only to the right. The periodic horizontal bands corresponding to the same hue indicate that $z \mapsto \exp z$ is periodic in the imaginary part of $z$ .

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 $0$ : $\frac{d e^{z}}{d z} = e^{z} and e^{0} = 1 .$

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 differential is an entire function.

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

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.^[11]

The complex logarithm is a right-inverse function of the complex exponential: $e^{\log z} = z .$ However, since the complex logarithm is a multivalued function, one has $\log e^{z} = {z + 2 i k π ∣ k \in ℤ},$ and it is difficult to define the complex exponential from the complex logarithm. On the opposite, this is the complex logarithm that is often defined from the complex exponential.

The complex exponential has the following properties: $\frac{1}{e^{z}} = e^{- z}$ and $e^{z} \neq 0 for every z \in ℂ .$ It is periodic function of period $2 i π$ ; that is $e^{z + 2 i k π} = e^{z} for every k \in ℤ .$ This results from Euler's identity $e^{i π} = - 1$ and the functional identity.

The complex conjugate of the complex exponential is $\overline{e^{z}} = e^{\overline{z}} .$ Its modulus is $| e^{z} | = e^{ℜ (z)},$ where $ℜ (z)$ denotes the real part of $z$ .

Relationship with trigonometry

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

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

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

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.^[12]

Plots

3D plots of real part, imaginary part, and modulus of the exponential function
$z = Re(e x + iy)$
$z = Im(e x + iy)$
$z = | e x + iy |$

Considering the complex exponential function as a function involving four real variables: $v + i w = \exp (x + i y)$ the graph of the exponential function is a two-dimensional surface curving through four dimensions.

Starting with a color-coded portion of the $x y$ domain, the following are depictions of the graph as variously projected into two or three dimensions.

Graphs of the complex exponential function
Checker board key:
$x > 0 : green$
$x < 0 : red$
$y > 0 : yellow$
$y < 0 : blue$
Projection onto the range complex plane (V/W). Compare to the next, perspective picture.
$Projection into the x {\displaystyle x} , v {\displaystyle v} , and w {\displaystyle w} dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image).$

Projection into the $x$ , $v$ , and $w$ dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image)
$Projection into the y {\displaystyle y} , v {\displaystyle v} , and w {\displaystyle w} dimensions, producing a spiral shape. ( y {\displaystyle y} range extended to ±2π, again as 2-D perspective image).$

Projection into the $y$ , $v$ , and $w$ dimensions, producing a spiral shape ( $y$ range extended to ±2 $π$ , again as 2-D perspective image)

The second image shows how the domain complex plane is mapped into the range complex plane:

zero is mapped to 1
the real $x$ axis is mapped to the positive real $v$ axis
the imaginary $y$ axis is wrapped around the unit circle at a constant angular rate
values with negative real parts are mapped inside the unit circle
values with positive real parts are mapped outside of the unit circle
values with a constant real part are mapped to circles centered at zero
values with a constant imaginary part are mapped to rays extending from zero

The third and fourth images show how the graph in the second image extends into one of the other two dimensions not shown in the second image.

The third image shows the graph extended along the real $x$ axis. It shows the graph is a surface of revolution about the $x$ axis of the graph of the real exponential function, producing a horn or funnel shape.

The fourth image shows the graph extended along the imaginary $y$ axis. It shows that the graph's surface for positive and negative $y$ values doesn't really meet along the negative real $v$ axis, but instead forms a spiral surface about the $y$ axis. Because its $y$ values have been extended to $\pm2 π$ , this image also better depicts the 2π periodicity in the imaginary $y$ value.

Matrices and Banach algebras

The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra $B$ . In this setting, $e 0 = 1$ , and $e x$ is invertible with inverse $e - x$ for any $x$ in $B$ . If $xy = yx$ , then $e x + y = e x e y$ , but this identity can fail for noncommuting $x$ and $y$ .

Some alternative definitions lead to the same function. For instance, $e x$ can be defined as $\lim_{n \to \infty} {(1 + \frac{x}{n})}^{n} .$

Or $e x$ can be defined as $f x (1)$ , where $f x : R \to B$ is the solution to the differential equation $⁠ df x / dt ⁠ (t) = x f x (t)$ , with initial condition $f x (0) = 1$ ; it follows that $f x (t) = e tx$ for every $t$ in $R$ .

Lie algebras

Given a Lie group $G$ and its associated Lie algebra $𝔤$ , the exponential map is a map $𝔤$ $\mapsto G$ satisfying similar properties. In fact, since $R$ is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie group $GL(n, R)$ of invertible $n \times n$ matrices has as Lie algebra $M(n, R)$ , the space of all $n \times n$ matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.

The identity $\exp (x + y) = \exp (x) \exp (y)$ can fail for Lie algebra elements $x$ and $y$ that do not commute; the Baker–Campbell–Hausdorff formula supplies the necessary correction terms.

Transcendency

The function $e z$ is a transcendental function, which means that it is not a root of a polynomial over the ring of the rational fractions $ℂ (z) .$

If $a 1, ..., a n$ are distinct complex numbers, then $e a 1 z, ..., e a n z$ are linearly independent over $ℂ (z)$ , and hence $e z$ is transcendental over $ℂ (z)$ .

Computation

The Taylor series definition above is generally efficient for computing (an approximation of) $e^{x}$ . However, when computing near the argument $x = 0$ , the result will be close to 1, and computing the value of the difference $e^{x} - 1$ with floating-point arithmetic may lead to the loss of (possibly all) significant figures, producing a large relative error, possibly even a meaningless result.

Following a proposal by William Kahan, it may thus be useful to have a dedicated routine, often called expm1, which computes $e x - 1$ directly, bypassing computation of $e x$ . For example, one may use the Taylor series: $e^{x} - 1 = x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \dots + \frac{x^{n}}{n!} + \dots .$

This was first implemented in 1979 in the Hewlett-Packard HP-41C calculator, and provided by several calculators,^[13]^[14] operating systems (for example Berkeley UNIX 4.3BSD^[15]), computer algebra systems, and programming languages (for example C99).^[16]

In addition to base $e$ , the IEEE 754-2008 standard defines similar exponential functions near 0 for base 2 and 10: $2^{x} - 1$ and $1 0^{x} - 1$ .

A similar approach has been used for the logarithm; see log1p.

An identity in terms of the hyperbolic tangent, $expm1 (x) = e^{x} - 1 = \frac{2 \tanh (x / 2)}{1 - \tanh (x / 2)},$ gives a high-precision value for small values of $x$ on systems that do not implement $expm1(x)$ .

Continued fractions

The exponential function can also be computed with continued fractions.

A continued fraction for $e x$ can be obtained via an identity of Euler: $e^{x} = 1 + \frac{x}{1 - \frac{x}{x + 2 - \frac{2 x}{x + 3 - \frac{3 x}{x + 4 - ⋱}}}}$

The following generalized continued fraction for $e z$ also received by Euler ^[17] converges more quickly:^[18] $e^{z} = 1 + \frac{2 z}{2 - z + \frac{z^{2}}{6 + \frac{z^{2}}{10 + \frac{z^{2}}{14 + ⋱}}}}$

or, by applying the substitution $z = ⁠ x / y ⁠$ : $e^{\frac{x}{y}} = 1 + \frac{2 x}{2 y - x + \frac{x^{2}}{6 y + \frac{x^{2}}{10 y + \frac{x^{2}}{14 y + ⋱}}}}$ with a special case for $z = 2$ : $e^{2} = 1 + \frac{4}{0 + \frac{2^{2}}{6 + \frac{2^{2}}{10 + \frac{2^{2}}{14 + ⋱}}}} = 7 + \frac{2}{5 + \frac{1}{7 + \frac{1}{9 + \frac{1}{11 + ⋱}}}}$

This formula also converges, though more slowly, for $z > 2$ . For example: $e^{3} = 1 + \frac{6}{- 1 + \frac{3^{2}}{6 + \frac{3^{2}}{10 + \frac{3^{2}}{14 + ⋱}}}} = 13 + \frac{54}{7 + \frac{9}{14 + \frac{9}{18 + \frac{9}{22 + ⋱}}}}$

Notes

References

↑ "Exponential Function Reference". https://www.mathsisfun.com/sets/function-exponential.html.
↑ Real and complex analysis (3rd ed.). New York: McGraw-Hill. 1987. p. 1. ISBN 978-0-07-054234-1. https://archive.org/details/RudinW.RealAndComplexAnalysis3e1987.
↑ ^3.0 ^3.1 Weisstein, Eric W.. "Exponential Function" (in en). https://mathworld.wolfram.com/ExponentialFunction.html.
↑ ^4.0 ^4.1 e: the Story of a Number. p. 156.
↑ G. Harnett, Calculus 1, 1998, Functions continued: "General exponential functions have the property that the ratio of two outputs depends only on the difference of inputs. The ratio of outputs for a unit change in input is the base."
↑ G. Harnett, Calculus 1, 1998; Functions continued / Exponentials & logarithms: "The ratio of outputs for a unit change in input is the base of a general exponential function."
↑ Serway, Raymond A.; Moses, Clement J.; Moyer, Curt A. (1989). Modern Physics. Fort Worth: Harcourt Brace Jovanovich. p. 384. ISBN 0-03-004844-3.
↑ Simmons, George F. (1972). Differential Equations with Applications and Historical Notes. New York: McGraw-Hill. p. 15.
↑ "McGraw-Hill Encyclopedia of Science & Technology". McGraw-Hill Encyclopedia of Science & Technology (10th ed.). New York: McGraw-Hill. 2007. ISBN 978-0-07-144143-8.
↑ O'Connor, John J.; Robertson, Edmund F., "Exponential function", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/HistTopics/e.html .
↑ Hille, Einar (1959). "The exponential function". Analytic Function Theory. 1. Waltham, MA: Blaisdell. § 6.1, Template:Pgs.
↑ Mathematical Analysis (2nd ed.). Reading, Mass.: Addison Wesley. 1974. pp. 19. ISBN 978-0-201-00288-1. https://archive.org/details/mathematicalanal00apos_530.
↑ HP 48G Series – Advanced User's Reference Manual (AUR) (4 ed.). Hewlett-Packard. December 1994. HP 00048-90136, 0-88698-01574-2. http://www.hpcalc.org/details.php?id=6036. Retrieved 2015-09-06.
↑ HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR) (2 ed.). Hewlett-Packard. 2009-07-14. HP F2228-90010. http://www.hpcalc.org/details.php?id=7141. Retrieved 2015-10-10. [1]
↑ "Chapter 10.2. Exponential near zero". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, UT, USA: Springer International Publishing AG. 2017-08-22. pp. 273–282. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. "Berkeley UNIX 4.3BSD introduced the expm1() function in 1987."
↑ "Computation of expm1 = exp(x)−1". Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing, University of Utah. 2002-07-09. http://www.math.utah.edu/~beebe/reports/expm1.pdf.
↑ A. N. Khovanski, The applications of continued fractions and their Generalisation to problemes in approximation theory,1963, Noordhoff, Groningen, The Netherlands
↑ "A.2.2 The exponential function.". Continued Fractions. Atlantis Studies in Mathematics. 1. 2008. p. 268. doi:10.2991/978-94-91216-37-4. ISBN 978-94-91216-37-4. https://link.springer.com/content/pdf/bbm%3A978-94-91216-37-4%2F1.

External links

Hazewinkel, Michiel, ed. (2001), "Exponential function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/e036910

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[1] "Exponential Function Reference". https://www.mathsisfun.com/sets/function-exponential.html.

[Rudin_1987-2] Real and complex analysis (3rd ed.). New York: McGraw-Hill. 1987. p. 1. ISBN 978-0-07-054234-1. https://archive.org/details/RudinW.RealAndComplexAnalysis3e1987.

[:0-3] 3.0 ^3.1 Weisstein, Eric W.. "Exponential Function" (in en). https://mathworld.wolfram.com/ExponentialFunction.html.

[Maor-4] 4.0 ^4.1 e: the Story of a Number. p. 156.

[5] G. Harnett, Calculus 1, 1998, Functions continued: "General exponential functions have the property that the ratio of two outputs depends only on the difference of inputs. The ratio of outputs for a unit change in input is the base."

[6] G. Harnett, Calculus 1, 1998; Functions continued / Exponentials & logarithms: "The ratio of outputs for a unit change in input is the base of a general exponential function."

[Serway-Moses-Moyer_1989-7] Serway, Raymond A.; Moses, Clement J.; Moyer, Curt A. (1989). Modern Physics. Fort Worth: Harcourt Brace Jovanovich. p. 384. ISBN 0-03-004844-3.

[Simmons_1972-8] Simmons, George F. (1972). Differential Equations with Applications and Historical Notes. New York: McGraw-Hill. p. 15.

[McGrawHill_2007-9] "McGraw-Hill Encyclopedia of Science & Technology". McGraw-Hill Encyclopedia of Science & Technology (10th ed.). New York: McGraw-Hill. 2007. ISBN 978-0-07-144143-8.

[O'Connor_2001-10] O'Connor, John J.; Robertson, Edmund F., "Exponential function", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/HistTopics/e.html .

[11] Hille, Einar (1959). "The exponential function". Analytic Function Theory. 1. Waltham, MA: Blaisdell. § 6.1, Template:Pgs.

[Apostol_1974-12] Mathematical Analysis (2nd ed.). Reading, Mass.: Addison Wesley. 1974. pp. 19. ISBN 978-0-201-00288-1. https://archive.org/details/mathematicalanal00apos_530.

[HP48_AUR-13] HP 48G Series – Advanced User's Reference Manual (AUR) (4 ed.). Hewlett-Packard. December 1994. HP 00048-90136, 0-88698-01574-2. http://www.hpcalc.org/details.php?id=6036. Retrieved 2015-09-06.

[HP50_AUR-14] HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR) (2 ed.). Hewlett-Packard. 2009-07-14. HP F2228-90010. http://www.hpcalc.org/details.php?id=7141. Retrieved 2015-10-10. [1]

[Beebe_2017-15] "Chapter 10.2. Exponential near zero". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, UT, USA: Springer International Publishing AG. 2017-08-22. pp. 273–282. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. "Berkeley UNIX 4.3BSD introduced the expm1() function in 1987."

[Beebe_2002-16] "Computation of expm1 = exp(x)−1". Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing, University of Utah. 2002-07-09. http://www.math.utah.edu/~beebe/reports/expm1.pdf.

[17] A. N. Khovanski, The applications of continued fractions and their Generalisation to problemes in approximation theory,1963, Noordhoff, Groningen, The Netherlands

[Lorentzen_2008-18] "A.2.2 The exponential function.". Continued Fractions. Atlantis Studies in Mathematics. 1. 2008. p. 268. doi:10.2991/978-94-91216-37-4. ISBN 978-94-91216-37-4. https://link.springer.com/content/pdf/bbm%3A978-94-91216-37-4%2F1.

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v t e Calculus
Precalculus	Binomial theorem Concave function Continuous function Factorial Finite difference Free variables and bound variables Fundamental theorem Graph of a function Linear function Mean value theorem Radian Rolle's theorem Secant Slope Tangent
Limits	Indeterminate form Limit of a function One-sided limit Limit of a sequence Orders of approximation (ε, δ)-definition of limit
Differential calculus	Chain rule Derivative Differential Differential equation Differential operator Implicit differentiation Inverse functions and differentiation L'Hôpital's rule Leibniz's rule Logarithmic derivative Mean value theorem Newton's method Notation Leibniz's notation Newton's notation Product rule Quotient rule Regiomontanus' angle maximization problem Related rates Simplest rules Constant factor rule in differentiation Linearity of differentiation Power rule Sum rule in differentiation Stationary point First derivative test Second derivative test Extreme value theorem Maxima and minima Taylor's theorem
Integral calculus	Antiderivative Arc length Constant of integration Differentiation under the integral sign Fundamental theorem of calculus Integral of secant cubed Integral of the secant function Integration by parts Integration by substitution Tangent half-angle substitution Partial fractions in integration Quadratic integral Proof that 22/7 exceeds π Simplest rules Constant factor rule in integration Linearity of integration Sum rule in integration Trapezoidal rule Trigonometric substitution
Vector calculus	Curl Directional derivative Divergence Divergence theorem Gradient Gradient theorem Green's theorem Laplacian Stokes' theorem
Multivariable calculus	Curvature Disc integration Divergence theorem Exterior Gabriel's Horn Geometric Hessian matrix Jacobian matrix and determinant Line integral Matrix Multiple integral Partial derivative Shell integration Surface integral Tensor Volume integral
Series	Abel's test Alternating Alternating series test Arithmetico-geometric sequence Binomial Cauchy condensation test Direct comparison test Dirichlet's test Euler–Maclaurin formula Fourier Geometric Harmonic Infinite Integral test for convergence Limit comparison test Maclaurin Power Ratio test Root test Taylor Term test
Special functions and numbers	Bernoulli numbers e (mathematical constant) Exponential function Natural logarithm Stirling's approximation
History of calculus	Adequality Brook Taylor Colin Maclaurin Generality of algebra Gottfried Wilhelm Leibniz Infinitesimal Infinitesimal calculus Isaac Newton Fluxion Law of Continuity Leonhard Euler Method of Fluxions The Method of Mechanical Theorems
Lists	Differentiation rules List of integrals of exponential functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of inverse trigonometric functions List of integrals of irrational functions List of integrals of logarithmic functions List of integrals of rational functions List of integrals of trigonometric functions List of limits List of mathematical symbols Lists of integrals

@@ Line 1: / Line 1: @@
 {{Short description|Mathematical function, denoted exp(x) or e^x}}
+{{About|the function {{math|{{var|f}}({{var|x}}) {{=}} {{var|e}}{{sup|{{var|x}}}}}} and its generalizations|functions of the form {{math|{{var|f}}({{var|x}}) {{=}} {{var|x}}{{sup|{{var|r}}}}}}|Power function|the bivariate function {{math|{{var|f}}({{var|x}},{{var|y}}) {{=}} {{var|x}}{{sup|{{var|y}}}}}}|Exponentiation|the representation of scientific numbers|E notation}}
 {{Infobox mathematical function
 | name = Exponential
 | image = Image:exp.svg
-| imagealt = The natural exponential function along part of the real axis
+| imagealt = Graph of the exponential function
-| caption = The natural exponential function along part of the real axis
+| caption = Graph of the exponential function
 | general_definition = <math>\exp z = e^{z}</math>
 | motivation_of_creation =
@@ Line 12: / Line 13: @@
 | zero = 1
 | vr1 = 1
-| f1 = ''e''
+| f1 = [[Euler's number|{{math|''e''}}]]
 | fixed = [[Lambert W function|{{math|−''W''{{sub|''n''}}(−1)}}]] for <math>n \in \mathbb{Z}</math>
 | reciprocal = <math>\exp(-z)</math>
 | inverse = [[Natural logarithm]], [[Complex logarithm]]
-| derivative = <math>\exp' z = \exp z</math>
+| derivative = <math>\exp'\! z = \exp z</math>
 | antiderivative = <math>\int \exp z\,dz = \exp z + C</math>
 | taylor_series = <math>\exp z = \sum_{n=0}^\infty\frac{z^n}{n!}</math>
 }}
-[[Image:Exponenciala priklad.png|thumb|200px|right|Exponential functions with bases 2 and 1/2]]
+In [[Mathematics|mathematics]], the '''exponential function''' is the unique [[Real function|real function]] which maps [[0|zero]] to [[1|one]] and has a [[Derivative|derivative]] everywhere equal to its value. The exponential of a variable {{tmath|x}} is denoted {{tmath|\exp x}} or {{tmath|e^x}}, with the two notations used interchangeably. It is called ''exponential'' because its argument can be seen as an exponent to which a constant [[e (mathematical constant)|number {{math|''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''' is a mathematical [[Function (mathematics)|function]] denoted by <math>f(x)=\exp(x)</math> or <math>e^x</math> (where the argument {{mvar|x}} is written as an [[Exponentiation|exponent]]). Unless otherwise specified, the term generally refers to the positive-valued function of a [[Real number|real variable]], although it can be extended to the [[Complex number|complex numbers]] or generalized to other mathematical objects like matrices or [[Lie algebra]]s. The exponential function originated from the operation of [[Exponentiation|taking powers]] of a number (repeated multiplication), but [[Characterizations of the exponential function|various modern definitions]] allow it to be rigorously extended to all real arguments <math>x</math>, including [[Irrational number|irrational numbers]]. Its ubiquitous occurrence in [[Pure mathematics|pure]] and [[Applied mathematics|applied mathematics]] led mathematician [[Biography:Walter Rudin|Walter Rudin]] to consider the exponential function to be "the most important function in mathematics".<ref name="Rudin_1987"/>
+The exponential function converts sums to products: it maps the [[Additive identity|additive identity]] {{math|0}} to the multiplicative identity {{math|1}}, and the exponential of a sum is equal to the product of separate exponentials, {{tmath|1=\exp(x + y) = \exp x \cdot \exp y }}. Its [[Inverse function|inverse function]], the [[Natural logarithm|natural logarithm]], {{tmath|\ln}} or {{tmath|\log}}, converts products to sums: {{tmath|1= \ln(x\cdot y) = \ln x + \ln y}}.
-The functions <math>f(x) = b^x</math> for positive real numbers <math>b</math> are also known as '''exponential functions''', and satisfy the exponentiation [[Identity (mathematics)|identity]]:<math display="block">b^{x+y} = b^x b^y \text{ for all } x,y\in\mathbb{R}.</math>This implies <math>b^n= b\times\cdots\times b</math> (with <math>n</math> factors) for positive integers <math>n</math>, where <math>b=b^1</math>, relating exponential functions to the elementary notion of exponentiation. The natural base <math>e = \exp(1)=2.71828\ldots</math> is a ubiquitous [[Mathematical constant|mathematical constant]] called [[E (mathematical constant)|Euler's number]]. To distinguish it, <math>\exp(x)=e^x</math> is called '''''the'' exponential function''' or the '''natural exponential function''': it is the unique real-valued function of a real variable whose derivative is itself and whose value at {{math|0}} is {{math|1}}: <blockquote><math>\exp'(x)=\exp(x)</math> for all <math>x\in \R</math>,  and  <math>\exp(0)=1.</math></blockquote>The relation <math>b^x = e^{x\ln b}</math> for <math>b>0</math> and real or complex <math>x</math> allows general exponential functions to be expressed in terms of the natural exponential.
+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 {{tmath|1=f(x) = b^x}}, which is [[Exponentiation|exponentiation]] with a fixed base {{tmath|b}}. More generally, and especially in applications, functions of the general form {{tmath|1=f(x) = ab^x}} are also called exponential functions. They [[Exponential growth|grow]] or [[Exponential decay|decay]] exponentially in that the rate that {{tmath|f(x)}} changes when {{tmath|x}} is increased is ''proportional'' to the current value of {{tmath|f(x)}}.
-More generally, especially in applied settings, any function <math>f:\mathbb{R}\to\mathbb{R}</math> defined by
+The exponential function can be generalized to accept [[Complex number|complex number]]s as arguments. This reveals relations between multiplication of complex numbers, rotations in the [[Complex plane|complex plane]], and [[Trigonometry|trigonometry]]. [[Euler's formula]] {{tmath|1= \exp i\theta = \cos\theta + i\sin\theta}} expresses and summarizes these relations.
-<math display="block">f(x)=c e^{ax}=c b^{kx}, \text{ with }k=a/ \ln b,\ a\neq 0,\ b ,c>0</math>
+The exponential function can be even further generalized to accept other types of arguments, such as matrices and elements of [[Exponential map (Lie theory)|Lie algebras]].
-is also known as an exponential function, as it solves the [[Initial value problem|initial value problem]] <math>f'=af,\ f(0)=c</math>, meaning its rate of change at each point is proportional to the value of the function at that point. This  behavior [[Mathematical model|models]] diverse phenomena in the biological, physical, and social sciences, for example the [[Exponential growth|unconstrained growth]] of a self-reproducing population, the [[Physics:Radioactive decay|decay of a radioactive element]], the [[Compound interest|compound interest]] accruing on a financial fund, or a [[Moore's law|growing body of manufacturing expertise]].
+==Graph==
+The [[Graph of a function|graph]] of <math>y=e^x</math> is upward-sloping, and increases faster than every power of {{tmath|x}}.<ref>{{Cite web|title=Exponential Function Reference|url=https://www.mathsisfun.com/sets/function-exponential.html|access-date=2020-08-28|website=www.mathsisfun.com}}</ref> The graph always lies above the {{mvar|x}}-axis, but becomes arbitrarily close to it for large negative {{mvar|x}}; thus, the {{mvar|x}}-axis is a horizontal [[Asymptote|asymptote]]. The equation <math>\tfrac{d}{dx}e^x = e^x</math> means that the [[Slope|slope]] of the [[Tangent|tangent]] to the graph at each point is equal to its height (its {{mvar|y}}-coordinate) at that point.
-The real exponential function can also be defined as a [[Taylor series|power series]], which is readily extended to complex arguments to define the complex exponential function <math>\exp:\mathbb{C}\to\mathbb{C}</math>. This function takes on all complex values except for 0 and is closely related to the complex [[Trigonometric functions|trigonometric functions]], as shown by [[Euler's formula]]: <blockquote><math>e^{x+iy} = e^x\cos(y) \,+\, i \,e^x\sin(y).</math> </blockquote>Motivated by its more abstract properties and characterizations, the exponential function can be generalized to much larger contexts such as [[Matrix exponential|square matrices]] and [[Exponential map (Lie theory)|Lie groups]]. Even further, the differential equation definition can be generalized to a [[Exponential map (Riemannian geometry)|Riemannian manifold]].
+==Definitions and fundamental properties==
+{{see also|Characterizations of the exponential function}}
+There are several equivalent definitions of the exponential function, although of very different nature.
-The real exponential function is a [[Bijection|bijection]] from <math>\mathbb{R}</math> to the interval <math>(0,\infty)</math>.<ref>{{cite book |last1=Meier |first1=John |last2=Smith |first2=Derek |title=Exploring Mathematics |date=7 August 2017 |publisher=Cambridge University Press |isbn=978-1-107-12898-9 |page=167 }}</ref> Its [[Inverse function|inverse function]] is the [[Natural logarithm|natural logarithm]], denoted <math>\ln</math>,<ref name="lnx" group="nb" /> <math>\log</math>,<ref name="Logx" group="nb" /> or <math>\log_e</math>, and some old texts<ref name="Durell_1911" /> called it the ''antilogarithm''.
+===Differential equation===
-==Graph==
+[[Image:Exp tangent.svg|thumb|right |The derivative of the exponential function is equal to the value of the function. Since the derivative is the [[Slope|slope]] of the tangent, this implies that all green [[Right triangle|right triangle]]s have a base length of 1.]]
-The [[Graph of a function|graph]] of <math>y=e^x</math> is upward-sloping, and increases faster as {{mvar|x}} increases.<ref>{{Cite web|title=Exponential Function Reference|url=https://www.mathsisfun.com/sets/function-exponential.html|access-date=2020-08-28|website=www.mathsisfun.com}}</ref> The graph always lies above the {{mvar|x}}-axis, but becomes arbitrarily close to it for large negative {{mvar|x}}; thus, the {{mvar|x}}-axis is a horizontal [[Asymptote|asymptote]]. The equation <math>\tfrac{d}{dx}e^x = e^x</math> means that the [[Slope|slope]] of the [[Tangent|tangent]] to the graph at each point is equal to its {{mvar|y}}-coordinate at that point.
-==Relation to more general exponential functions==
+One of the simplest definitions is: The ''exponential function'' is the ''unique'' [[Differentiable function|differentiable function]] that equals its [[Derivative|derivative]], and takes the value {{math|1}} for the value {{math|0}} of its variable.
-The exponential function <math>f(x) = e^x</math> is sometimes called the ''natural exponential function'' for distinguishing it from the other exponential functions. The study of any exponential function can easily be reduced to that of the natural exponential function, since per definition, for positive {{mvar|b}},
-<math display="block"> b^x \mathrel{\stackrel{\text{def}}{=}} e^{x\ln b} </math>
-As functions of a real variable, exponential functions are uniquely [[Characterization (mathematics)|characterized]] by the fact that the [[Derivative|derivative]] of such a function is [[Proportionality (mathematics)#Direct proportionality|directly proportional]] to the value of the function. The constant of proportionality of this relationship is the [[Natural logarithm|natural logarithm]] of the base {{mvar|b}}:
+This "conceptual" definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.
-<math display="block">\frac{d}{dx} b^x = \frac{d}{dx} e^{x\ln (b)} = e^{x\ln (b)} \ln (b) = b^x \ln (b).</math>
-For {{math|''b'' > 1}}, the function <math>b^x</math> is increasing (as depicted for {{math|1=''b'' = ''e''}} and {{math|1=''b'' = 2}}), because <math>\ln b>0</math> makes the derivative always positive; this is often referred to as [[Exponential growth|exponential growth]]. For positive {{math|''b'' < 1}}, the function is decreasing (as depicted for {{math|1=''b'' = {{sfrac|1|2}}}}); this is often referred to as [[Exponential decay|exponential decay]].  For {{math|1=''b'' = 1}}, the function is constant.
+''Uniqueness: ''If {{tmath|f(x)}} and {{tmath|g(x)}} are two functions satisfying the above definition, then the derivative of {{tmath|f/g}} is zero everywhere because of the [[Quotient rule|quotient rule]]. It follows that {{tmath|f/g}} is constant; this constant is {{math|1}} since {{tmath|1=f(0) = g(0)=1}}.
-Euler's number {{math|1=''e'' = 2.71828...}}<ref>{{Cite OEIS|A001113|Decimal expansion of e}}</ref> is the unique base for which the constant of proportionality is 1, since <math>\ln(e) = 1</math>, so that the function is its own derivative:
+''Existence'' is proved in each of the two following sections.
-<math display="block">\frac{d}{dx} e^x = e^x \ln (e) = e^x.</math>
-This function, also denoted as {{math|exp ''x''}}, is called the "natural exponential function",<ref name="Goldstein_2006"/><ref name="Courant_1996"/> or simply "the exponential function". Since any exponential function defined by <math>f(x)=b^x</math> can be written in terms of the natural exponential as <math>b^x = e^{x\ln b}</math>, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. The natural exponential is hence denoted by
+===Inverse of natural logarithm===
-<math display="block">x\mapsto e^x</math> or <math display="block">x\mapsto \exp x.</math>
+''The exponential function is the [[Inverse function|inverse function]] of the [[Natural logarithm|natural logarithm]].'' The [[Inverse function theorem|inverse function theorem]] implies that the natural logarithm has an inverse function,  that satisfies the above definition. This is a first proof of existence. Therefore, one has
+:<math>\begin{align}
+\ln (\exp x)&=x\\
+\exp(\ln y)&=y
+\end{align}</math>
+for every [[Real number|real number]] <math>x</math> and every positive real number <math>y.</math>
-The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent is more complicated and harder to read in a small font.
+===Power series===
-For real numbers {{mvar|c}} and {{mvar|d}}, a function of the form <math>f(x) = a b^{cx + d}</math> is also an exponential function, since it can be rewritten as
+''The exponential function is the sum of the [[Power series|power series]]''<ref name="Rudin_1987"/><ref name=":0">{{Cite web|last=Weisstein| first=Eric W.|title=Exponential Function|url=https://mathworld.wolfram.com/ExponentialFunction.html|access-date=2020-08-28| website=mathworld.wolfram.com|language=en}}</ref>
-<math display="block">a b^{c x + d} = \left(a b^d\right) \left(b^c\right)^x.</math>
+<math display=block>
+\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}</math>
+[[Image:Exp series.gif|right|thumb|The exponential function (in blue), and the sum of the first {{math|''n'' + 1}} terms of its power series (in red)]]
+where <math>n!</math> is the [[Factorial|factorial]] of {{mvar|n}} (the product of the {{mvar|n}} first positive integers). This series is absolutely convergent for every <math>x</math> per the [[Ratio test|ratio test]]. So, the derivative of the sum can be computed by term-by-term differentiation, and this shows that the sum of the series satisfies the above definition. This is a second existence proof, and shows, as a byproduct, that the exponential function is defined for every {{tmath|x}}, and is everywhere the sum of its [[Maclaurin series]].
-==Formal definition==
+===Functional equation===
-{{main|Characterizations of the exponential function}}
+''The exponential satisfies the [[Functional equation|functional equation]]:''
-[[Image:Exp series.gif|right|thumb|The exponential function (in blue), and the sum of the first {{math|''n'' + 1}} terms of its power series (in red).]]
+<math display=block>\exp(x+y)= \exp(x)\cdot \exp(y).</math>
+This results from the uniqueness and the fact that the function
+<math> f(x)=\exp(x+y)/\exp(y)</math> satisfies the above definition.
-The real exponential function <math>\exp:\mathbb{R}\to\mathbb{R}</math> can be characterized in a variety of equivalent ways. It is commonly defined by the following [[Power series|power series]]:<ref name="Rudin_1987"/><ref name=":0">{{Cite web|last=Weisstein| first=Eric W.|title=Exponential Function|url=https://mathworld.wolfram.com/ExponentialFunction.html|access-date=2020-08-28| website=mathworld.wolfram.com|language=en}}</ref>
+It can be proved that a function that satisfies this functional equation has the form {{tmath|x \mapsto \exp(cx)}} if it is either [[Continuous function|continuous]] or [[Monotonic function|monotonic]]. It is thus [[Differentiable function|differentiable]], and equals the exponential function if its derivative at {{math|0}} is {{math|1}}.
-<math display="block">\exp x := \sum_{k = 0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots</math>
-Since the [[Radius of convergence|radius of convergence]] of this power series is infinite, this definition is, in fact, applicable to all complex numbers; see {{Section link||Complex plane}} for the extension of <math>\exp x</math> to the complex plane. Using the power series, the constant {{mvar|e}} can be defined as <math display="inline">e = \exp 1 = \sum_{k=0}^\infty(1/k!)\,.</math>
+===Limit of integer powers===
+''The exponential function is the [[Limit (mathematics)|limit]], as the integer {{mvar|n}} goes to infinity,<ref name="Maor"/><ref name=":0" />
+<math display=block>\exp(x)=\lim_{n \to +\infty} \left(1+\frac xn\right)^n.</math>
+By continuity of the logarithm, this can be proved by taking logarithms and proving
+<math display=block>x=\lim_{n\to\infty}\ln \left(1+\frac xn\right)^n= \lim_{n\to\infty}n\ln \left(1+\frac xn\right),</math>
+for example with [[Taylor's theorem]].
-The term-by-term differentiation of this power series reveals that <math display="inline">\frac{d}{dx}\exp x = \exp x</math> for all real {{mvar|x}}, leading to another common characterization of <math>\exp x</math> as the unique solution of the [[Differential equation|differential equation]]
+===Properties===
-<math display="block">y'(x) = y(x)</math>
+''[[Multiplicative inverse|Reciprocal]]:'' The functional equation implies {{tmath|1=e^x e^{-x}=1}}. Therefore {{tmath|e^x \ne 0}} for every {{tmath|x}} and
-that satisfies the initial condition <math>y(0) = 1.</math>
+<math display=block>\frac 1{e^x}=e^{-x}.</math>
-Based on this characterization, the [[Chain rule|chain rule]] shows that its inverse function, the [[Natural logarithm|natural logarithm]], satisfies <math display="inline">\frac{d}{dy}\ln y = 1/y</math> for <math>y > 0,</math> or <math display="inline">\ln y = \int_1^y \frac{dt}{t}\,.</math> This relationship leads to a less common definition of the real exponential function <math>\exp(x)</math> as the solution <math>y</math> to the equation
+''Positiveness:'' {{tmath|e^x>0}} for every real number {{tmath|x}}. This results from the [[Intermediate value theorem|intermediate value theorem]], since {{tmath|1=e^0=1}} and, if one would have {{tmath|e^x<0}} for some {{tmath|x}}, there would be an {{tmath|y}} such that {{tmath|1=e^y=0}} between {{tmath|0}} and {{tmath|x}}. Since the exponential function equals its derivative, this implies that the exponential function is monotonically increasing.
-<math display="block">x = \int_1^y \frac{1}{t} \, dt.</math>
-Solving the [[Ordinary differential equation|ordinary differential equation]] <math>y'(x) = y(x)</math> with the [[Initial value problem|initial condition]] <math>y'(0)=1</math> using [[Euler method|Euler's method]] gives the product limit formula, valid for all complex values of <math>x</math>:<ref name="Maor"/><ref name=":0" />
+''Extension of [[Exponentiation|exponentiation]] to positive real bases:'' Let {{mvar|b}} be a positive real number. The exponential function and the natural logarithm being the inverse each of the other, one has <math>b=\exp(\ln b).</math> If {{mvar|n}} is an integer, the functional equation of the logarithm implies
-<math display="block">\exp x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n.</math>
+<math display=block>b^n=\exp(\ln b^n)= \exp(n\ln b).</math>
+Since the right-most expression is defined if {{mvar|n}} is any real number, this allows defining {{tmath|b^x}} for every positive real number {{mvar|b}} and every real number {{mvar|x}}:
+<math display=block>b^x=\exp(x\ln b).</math>
+In particular, if {{mvar|b}} is the Euler's number <math>e=\exp(1),</math> one has <math>\ln e=1</math> (inverse function) and thus <math display=block>e^x=\exp(x).</math> This shows the equivalence of the two notations for the exponential function.
-It can be shown that every [[Continuous function|continuous]], nonzero solution of the functional equation <math>f(x+y)=f(x)f(y)</math> for <math>f: \R \to \R</math> is an exponential function, <math>f(x) = e^{kx}</math> with <math>k\in\mathbb{R}.</math>
+==General exponential functions==
-==Overview==
+A function is commonly called ''an exponential function''{{mdash}}with an indefinite article{{mdash}}if it has the form {{tmath|x \mapsto b^x}}, that is, if it is obtained from [[Exponentiation|exponentiation]] by fixing the base and letting the ''exponent'' vary.
-[[File:Animation of exponential function.gif|thumb|The red curve is the exponential function.  The black horizontal lines show where it crosses the green vertical lines.]]
-The exponential function arises whenever a quantity [[Exponential growth|grows]] or [[Exponential decay|decays]] at a rate [[Proportionality (mathematics)|proportional]] to its current value.  One such situation is continuously compounded interest, and in fact it was this observation that led [[Biography:Jacob Bernoulli|Jacob Bernoulli]] in 1683<ref name="O'Connor_2001"/> to the number
-<math display="block">\lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^{n}</math>
-now known as {{math|''e''}}.  Later, in 1697, [[Biography:Johann Bernoulli|Johann Bernoulli]] studied the calculus of the exponential function.<ref name="O'Connor_2001"/>
-If a principal amount of 1 earns interest at an annual rate of {{math|''x''}} compounded monthly, then the interest earned each month is {{math|{{sfrac|''x''|12}}}} times the current value, so each month the total value is multiplied by {{math|(1 + {{sfrac|''x''|12}})}}, and the value at the end of the year is {{math|(1 + {{sfrac|''x''|12}})<sup>12</sup>}}.  If instead interest is compounded daily, this becomes {{math|(1 + {{sfrac|''x''|365}})<sup>365</sup>}}.  Letting the number of time intervals per year grow without bound leads to the [[Limit of a function|limit]] definition of the exponential function,
+More generally and especially in applied contexts, the term ''exponential function'' is commonly used for functions of the form {{tmath|1=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 {{tmath|a}}, and so, it is nonsensical to impose {{tmath|1=a=1}}.
-<math display="block">\exp x = \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^{n}</math>
-first given by [[Biography:Leonhard Euler|Leonhard Euler]].<ref name="Maor"/>
-This is one of a number of [[Characterizations of the exponential function|characterizations of the exponential function]]; others involve [[Series (mathematics)|series]] or [[Differential equation|differential equation]]s.
-From any of these definitions it can be shown that {{math|''e''{{sup|−''x''}}}} is the reciprocal of {{math|''e''{{sup|''x''}}}}.  For example from the differential equation definition, {{math|1=''e''{{sup|''x''}} ''e''{{sup|−''x''}} = 1}} when {{math|1=''x'' = 0}} and its derivative using the [[Product rule|product rule]] is {{math|1=''e''{{sup|''x''}} ''e''{{sup|−''x''}} − ''e''{{sup|''x''}} ''e''{{sup|−''x''}} = 0}} for all {{mvar|x}}, so {{math|1=''e''{{sup|''x''}} ''e''{{sup|−''x''}} = 1}} for all {{mvar|x}}.
+These most general exponential functions are the [[Differentiable function|differentiable function]]s that satisfy the following equivalent characterizations.
+* {{tmath|1=f(x) = ab^x}} for every {{tmath|x}} and some constants {{tmath|a}} and {{tmath|b>0}}.
+* {{tmath|1=f(x)=ae^{kx} }} for every {{tmath|x}} and some constants {{tmath|a}} and {{tmath|k}}.
+* The value of <math>f'(x)/f(x)</math>  is independent of <math>x</math>.
+* For every <math>d,</math> the value of <math>f(x+d)/f(x)</math> is independent of <math>x;</math> that is, <math display=block>\frac{f(x+d)}{f(x)}= \frac{f(y+d)}{f(y)}</math> for every {{mvar|x}}, {{mvar|y}}.<ref>G. Harnett, ''Calculus 1'', 1998, Functions continued:
+"General exponential functions have the property that the ratio of two outputs depends only on the difference of inputs. The ratio of outputs for a unit change in input is the base."</ref>
-From any of these definitions it can be shown that the exponential function obeys the basic [[Exponentiation|exponentiation]] identity.  For example from the power series definition,
+[[Image:Exponenciala priklad.png|thumb|200px|right|Exponential functions with bases 2 and 1/2]]
-<math display="block">\exp(x + y)
+The ''base'' of an exponential function is the ''base'' of the [[Exponentiation|exponentiation]] that appears in it when written as {{tmath|x\to ab^x}}, namely {{tmath|b}}.<ref>G. Harnett, ''Calculus 1'', 1998; Functions continued / Exponentials & logarithms: "The ratio of outputs for a unit change in input is the ''base'' of a general exponential function."</ref> The base is {{tmath|e^k}} in the second characterization, <math display=inline>\exp \frac{f'(x)}{f(x)}</math> in the third one, and <math display=inline>\left(\frac{f(x+d)}{f(x)}\right)^{1/d}</math> in the last one.
-= \sum_{m=0}^{\infty} \frac{(x+y)^m}{m!}
-= \sum_{m=0}^{\infty} \sum_{k=0}^m\frac{m!}{k! (m-k)!} \frac{x^k y^{m-k}}{m!}
-= \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{x^k y^n}{k! n!}
-= \exp x \cdot \exp y\,.</math>
-This justifies the notation {{math|''e''<sup>''x''</sup>}} for {{math|exp ''x''}}.
-The [[Derivative|derivative]] (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change ''proportional'' to the function itself (rather than equal to it) is expressible in terms of the exponential function. This function property leads to [[Exponential growth|exponential growth]] or [[Exponential decay|exponential decay]].
+===In applications===
+The last characterization is important in empirical sciences, as allowing a direct experimental test whether a function is an exponential function.
-The exponential function extends to an [[Entire function|entire function]] on the [[Complex plane|complex plane]]. [[Euler's formula]] relates its values at purely imaginary arguments to [[Trigonometric functions|trigonometric functions]].  The exponential function also has analogues for which the argument is a [[Matrix exponential|matrix]], or even an element of a [[Banach algebra]] or a [[Lie algebra]].
+Exponential [[Exponential growth|growth]] or [[Exponential decay|exponential decay]]{{mdash}}where the variable change is [[Proportionality (mathematics)|proportional]] to the variable value{{mdash}}are thus modeled with exponential functions. Examples are unlimited population growth leading to [[Unsolved:Malthusian catastrophe|Malthusian catastrophe]], [[Compound interest#Continuous compounding|continuously compounded interest]], and [[Physics:Radioactive decay|radioactive decay]].
-==Derivatives and differential equations==
+If the modeling function has the form {{tmath|x\mapsto ae^{kx},}} or, equivalently, is a solution of the differential equation {{tmath|1=y'=ky}}, the constant {{tmath|k}} is called, depending on the context, the ''decay constant'', ''disintegration constant'',<ref name="Serway-Moses-Moyer_1989" /> ''rate constant'',<ref name="Simmons_1972" /> or ''transformation constant''.<ref name="McGrawHill_2007" />
-[[Image:Exp tangent.svg|thumb|250px|right |The derivative of the exponential function is equal to the value of the function. From any point {{math|''P''}} on the curve (blue), let a tangent line (red), and a vertical line (green) with height {{math|''h''}} be drawn, forming a right triangle with a base {{math|''b''}} on the {{math|''x''}}-axis. Since the slope of the red tangent line (the derivative) at {{math|''P''}} is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function, {{math|''h''}} must be equal to the ratio of {{math|''h''}} to {{math|''b''}}. Therefore, the base {{mvar|b}} must always be&nbsp;1.]]
-The importance of the exponential function in mathematics and the sciences stems mainly from its property as the unique function which is equal to its derivative and is equal to 1 when {{math|1=''x'' = 0}}. That is,
+===Equivalence proof===
-<math display="block">\frac{d}{dx}e^x = e^x \quad\text{and}\quad e^0=1.</math>
+For proving the equivalence of the above properties, one can proceed as follows.
-Functions of the form {{math|''ce''<sup>''x''</sup>}} for constant {{math|''c''}} are the only functions that are equal to their derivative (by the [[Picard–Lindelöf theorem]]). Other ways of saying the same thing include:
+The two first characterizations are equivalent, since, if {{tmath|1=b=e^k}} and {{tmath|1= k=\ln b}}, one has
-* The slope of the graph at any point is the height of the function at that point.
+<math display=block>e^{kx}= (e^k)^x= b^x.</math>
-* The rate of increase of the function at {{math|''x''}} is equal to the value of the function at {{math|''x''}}.
+The basic properties of the exponential function (derivative and functional equation) implies immediately the third and the last condition.
-* The function solves the [[Differential equation|differential equation]] {{math|1=''y''′ = ''y''}}.
-* {{math|exp}} is a [[Fixed point (mathematics)|fixed point]] of derivative as a [[Functional (mathematics)|functional]].
-If a variable's growth or decay rate is [[Proportionality (mathematics)|proportional]] to its size—as is the case in unlimited population growth (see [[Unsolved:Malthusian catastrophe|Malthusian catastrophe]]), continuously compounded [[Finance:Interest|interest]], or [[Physics:Radioactive decay|radioactive decay]]—then the variable can be written as a constant times an exponential function of time.  Explicitly for any real constant {{math|''k''}}, a function {{math|''f'': '''R''' → '''R'''}} satisfies {{math|1=''f''′ = ''kf''}} if and only if {{math|1=''f''(''x'') = ''ce''<sup>''kx''</sup>}} for some constant {{math|''c''}}. The constant ''k'' is called the '''decay constant''', '''disintegration constant''',<ref name="Serway-Moses-Moyer_1989"/> '''rate constant''',<ref name="Simmons_1972"/> or '''transformation constant'''.<ref name="McGrawHill_2007"/>
+Suppose that the third condition is verified, and let {{tmath|k}} be the constant value of <math>f'(x)/f(x).</math> Since <math display = inline>\frac {\partial e^{kx}}{\partial x}=ke^{kx},</math> the [[Quotient rule|quotient rule]] for derivation
+implies that
+<math display=block>\frac \partial{\partial x}\,\frac{f(x)}{e^{kx}}=0,</math> and thus that there is a constant {{tmath|a}} such that <math>f(x)=ae^{kx}.</math>
-Furthermore, for any differentiable function {{math|''f''}}, we find, by the [[Chain rule|chain rule]]:
+If the last condition is verified, let <math display=inline>\varphi(d)=f(x+d)/f(x),</math> which is independent of {{tmath|x}}. Using {{tmath|1=\varphi (0)=1}}, one gets
-<math display="block">\frac{d}{dx} e^{f(x)} = f'(x)e^{f(x)}.</math>
+<math display=block>\frac{f(x+d)-f(x)}{d} = f(x)\,\frac{\varphi(d)-\varphi(0)}{d}. </math>
+Taking the limit when {{tmath|d}} tends to zero, one gets that the third condition is verified with {{tmath|1=k=\varphi'(0)}}. It follows therefore that {{tmath|1=f(x)= ae^{kx} }} for some {{tmath|a,}} and {{tmath|1=\varphi(d)= e^{kd}.}} As a byproduct, one gets that
+<math display=block>\left(\frac{f(x+d)}{f(x)}\right)^{1/d}=e^k</math>
+is independent of both {{tmath|x}} and {{tmath|d}}.
-==Continued fractions for {{small|{{math|''e''<sup>''x''</sup>}}}}==
+==Compound interest==
-A [[Continued fraction|continued fraction]] for {{math|''e''<sup>''x''</sup>}} can be obtained via [[Euler's continued fraction formula|an identity of Euler]]:
-<math display="block"> e^x = 1 + \cfrac{x}{1 - \cfrac{x}{x + 2 - \cfrac{2x}{x + 3 - \cfrac{3x}{x + 4 - \ddots}}}}</math>
-The following [[Generalized continued fraction|generalized continued fraction]] for {{math|''e''<sup>''z''</sup>}} converges more quickly:<ref name="Lorentzen_2008"/>
+The earliest occurrence of the exponential function was in [[Biography:Jacob Bernoulli|Jacob Bernoulli]]'s study of [[Compound interest|compound interest]]s in 1683.<ref name="O'Connor_2001"/>
-<math display="block"> e^z = 1 + \cfrac{2z}{2 - z + \cfrac{z^2}{6 + \cfrac{z^2}{10 + \cfrac{z^2}{14 + \ddots}}}}</math>
+This is this study that led Bernoulli to consider the number
+<math display="block">\lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^{n}</math>
+now known as Euler's number and denoted {{tmath|e}}.
-or, by applying the substitution {{math|1=''z'' = {{sfrac|''x''|''y''}}}}:
+The exponential function is involved as follows in the computation of [[Compound interest#Continuous compounding|continuously compounded interests]].
-<math display="block">  e^\frac{x}{y} = 1 + \cfrac{2x}{2y - x + \cfrac{x^2} {6y + \cfrac{x^2} {10y + \cfrac{x^2} {14y + \ddots}}}}</math>
-with a special case for {{math|1=''z'' = 2}}:
-<math display="block">  e^2 = 1 + \cfrac{4}{0 + \cfrac{2^2}{6 + \cfrac{2^2}{10 + \cfrac{2^2}{14 + \ddots\,}}}} = 7 + \cfrac{2}{5 + \cfrac{1}{7 + \cfrac{1}{9 + \cfrac{1}{11 + \ddots\,}}}}</math>
-This formula also converges, though more slowly, for {{math|''z'' > 2}}. For example:
+If a principal amount of 1 earns interest at an annual rate of {{math|''x''}} compounded monthly, then the interest earned each month is {{math|{{sfrac|''x''|12}}}} times the current value, so each month the total value is multiplied by {{math|(1 + {{sfrac|''x''|12}})}}, and the value at the end of the year is {{math|(1 + {{sfrac|''x''|12}})<sup>12</sup>}}.  If instead interest is compounded daily, this becomes {{math|(1 + {{sfrac|''x''|365}})<sup>365</sup>}}.  Letting the number of time intervals per year grow without bound leads to the [[Limit of a function|limit]] definition of the exponential function,
-<math display="block"> e^3 = 1 + \cfrac{6}{-1 + \cfrac{3^2}{6 + \cfrac{3^2}{10 + \cfrac{3^2}{14 + \ddots\,}}}} = 13 + \cfrac{54}{7 + \cfrac{9}{14 + \cfrac{9}{18 + \cfrac{9}{22 + \ddots\,}}}}</math>
+<math display="block">\exp x = \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^{n}</math>
+first given by [[Biography:Leonhard Euler|Leonhard Euler]].<ref name="Maor"/>
-=={{anchor|On the complex plane}}Complex plane==
-[[File:The exponential function e^z plotted in the complex plane from -2-2i to 2+2i.svg|alt=The exponential function e^z plotted in the complex plane from -2-2i to 2+2i|thumb|The exponential function e^z plotted in the complex plane from -2-2i to 2+2i]]<!-- This section is linked from [[Complex plane]] -->
-[[Image:Exp-complex-cplot.svg|thumb|right|A [[Domain coloring|complex plot]] of <math>z\mapsto\exp z</math>, with the [[Argument (complex analysis)|argument]] <math>\operatorname{Arg}\exp z</math> represented by varying hue. The transition from dark to light colors shows that <math>\left|\exp z\right|</math> is increasing only to the right. The periodic horizontal bands corresponding to the same hue indicate that <math>z\mapsto\exp z</math> is [[Periodic function|periodic]] in the imaginary part of <math>z</math>.]]
-As in the [[Real number|real]] case, the exponential function can be defined on the [[Complex plane|complex plane]] in several equivalent forms.
+==Differential equations==
+{{main|Linear differential equation}}
+Exponential functions occur very often in solutions of [[Differential equation|differential equation]]s.
-The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one:
+The exponential functions can be defined as solutions of [[Differential equation|differential equation]]s. Indeed, the exponential function is a solution of the simplest possible differential equation, namely {{tmath|1=y'=y}}. Every other exponential function, of the form {{tmath|1=y=ab^x}}, is a solution of the differential equation {{tmath|1=y'=ky}}, and every solution of this differential equation has this form.
-<math display="block">\exp z := \sum_{k = 0}^\infty\frac{z^k}{k!} </math>
-Alternatively, the complex exponential function may be defined by modelling the limit definition for real arguments, but with the real variable replaced by a complex one:
+The solutions of an equation of the form
-<math display="block">\exp z := \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n </math>
+<math display=block>y'+ky=f(x)</math>
+involve exponential functions in a more sophisticated way, since they have the form
+<math display=block>y=ce^{-kx}+e^{-kx}\int f(x)e^{kx}dx,</math>
+where {{tmath|c}} is an arbitrary constant and the integral denotes any [[Antiderivative|antiderivative]] of its argument.
-For the power series definition, term-wise multiplication of two copies of this power series in the [[Cauchy product|Cauchy]] sense, permitted by [[Cauchy product|Mertens' theorem]], shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments:
+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.
-<math display="block">\exp(w+z)=\exp w\exp z \text { for all } w,z\in\mathbb{C}</math>
-The definition of the complex exponential function in turn leads to the appropriate definitions extending the [[Trigonometric functions|trigonometric functions]] to complex arguments.
+==Complex exponential==
+{{anchor|On the complex plane|Complex plane}}
+[[File:The exponential function e^z plotted in the complex plane from -2-2i to 2+2i.svg|alt=The exponential function {{math|''e''{{isup|''z''}}}} plotted in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}|thumb|The exponential function {{math|''e''{{isup|''z''}}}} plotted in the complex plane from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}}]]
+[[Image:Exp-complex-cplot.svg|thumb|right|A [[Domain coloring|complex plot]] of <math>z\mapsto\exp z</math>, with the [[Argument (complex analysis)|argument]] <math>\operatorname{Arg}\exp z</math> represented by varying hue. The transition from dark to light colors shows that <math>\left|\exp z\right|</math> is increasing only to the right. The periodic horizontal bands corresponding to the same hue indicate that <math>z\mapsto\exp z</math> is [[Periodic function|periodic]] in the imaginary part of <math>z</math>.]]
-In particular, when {{math|1=''z'' = ''it''}} ({{mvar|t}} real), the series definition yields the expansion
+The exponential function can be naturally extended to a complex function, which is a function with the [[Complex number|complex number]]s as [[Domain of a function|domain]] and [[Codomain|codomain]], such that its [[Restriction (mathematics)|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 {{tmath|e^z}} or {{tmath|\exp(z)}}. For distinguishing the complex case from the real one, the extended function is also called '''complex exponential function''' or simply '''complex exponential'''.
-<math display="block">\exp(it) = \left( 1-\frac{t^2}{2!}+\frac{t^4}{4!}-\frac{t^6}{6!}+\cdots \right) + i\left(t - \frac{t^3}{3!} + \frac{t^5}{5!} - \frac{t^7}{7!}+\cdots\right).</math>
-In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series.  The real and imaginary parts of the above expression in fact correspond to the series expansions of {{math|cos ''t''}} and {{math|sin ''t''}}, respectively.
+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.
-This correspondence provides motivation for {{em|defining}} cosine and sine for all complex arguments in terms of <math>\exp(\pm iz)</math> and the equivalent power series:<ref name="Rudin_1976"/>
+The complex exponential function can be defined in several equivalent ways that are the same as in the real case.
-<math display="block">\begin{align}
-   & \cos z:= \frac{\exp(iz)+\exp(-iz)}{2} = \sum_{k=0}^\infty (-1)^k \frac{z^{2k}}{(2k)!}, \\[5pt]
-  \text{and } \quad & \sin z := \frac{\exp(iz)-\exp(-iz)}{2i} =\sum_{k=0}^\infty (-1)^k\frac{z^{2k+1}}{(2k+1)!}
- \end{align}</math>
-for all <math display=inline> z\in\mathbb{C}.</math>
+The ''complex exponential'' is the unique complex function that equals its complex derivative and takes the value {{tmath|1}} for the argument {{tmath|0}}:
+<math display="block">\frac{de^z}{dz}=e^z\quad\text{and}\quad e^0=1.</math>
-The functions {{math|exp}}, {{math|cos}}, and {{math|sin}} so defined have infinite [[Radius of convergence|radii of convergence]] by the [[Ratio test|ratio test]] and are therefore [[Entire function|entire function]]s (that is, [[Holomorphic function|holomorphic]] on <math>\mathbb{C}</math>).  The range of the exponential function is <math>\mathbb{C}\setminus \{0\}</math>, while the ranges of the complex sine and cosine functions are both <math>\mathbb{C}</math> in its entirety, in accord with [[Picard theorem|Picard's theorem]], which asserts that the range of a nonconstant entire function is either all of <math>\mathbb{C}</math>, or <math>\mathbb{C}</math> excluding one [[Lacunary value|lacunary value]].
+The ''complex exponential function'' is the sum of the [[Series (mathematics)|series]]
+<math display="block">e^z = \sum_{k = 0}^\infty\frac{z^k}{k!}.</math>
+This series is absolutely convergent for every complex number {{tmath|z}}. So, the complex differential is an [[Entire function|entire function]].
-These definitions for the exponential and trigonometric functions lead trivially to [[Euler's formula]]:
+The complex exponential function is the [[Limit (mathematics)|limit]]
-<math display="block">\exp(iz)=\cos z+i\sin z \text { for all } z\in\mathbb{C}.</math>
+<math display="block">e^z = \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n</math>
-We could alternatively define the complex exponential function based on this relationship.  If {{math|1=''z'' = ''x'' + ''iy''}}, where {{mvar|x}} and {{mvar|y}} are both real, then we could define its exponential as
+As with the real exponential function (see {{slink||Functional equation}} above), the complex exponential satisfies the functional equation
-<math display="block">\exp z = \exp(x+iy) :=  (\exp x)(\cos y + i \sin y)</math>
+<math display=block>\exp(z+w)= \exp(z)\cdot \exp(w).</math>
-where {{math|exp}}, {{math|cos}}, and {{math|sin}} on the right-hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means.<ref name="Apostol_1974"/>
+Among complex functions, it is the unique solution which is holomorphic at the point {{tmath|1= z = 0}} and takes the derivative {{tmath|1}} there.<ref>{{cite book |last=Hille |first=Einar |year=1959 |title=Analytic Function Theory |volume=1 |place=Waltham, MA |publisher=Blaisdell |chapter=The exponential function |at=§ 6.1, {{pgs|138–143}} }}</ref>
-For <math>t\in\R</math>, the relationship <math>\overline{\exp(it)}=\exp(-it)</math> holds, so that <math>\left|\exp(it)\right| = 1</math> for real <math>t</math> and <math>t \mapsto \exp(it)</math> maps the real line (mod {{math|2''π''}}) to the [[Unit circle|unit circle]] in the complex plane.  Moreover, going from <math>t = 0</math> to <math>t = t_0</math>, the curve defined by <math>\gamma(t)=\exp(it)</math> traces a segment of the unit circle of length
+The [[Complex logarithm|complex logarithm]] is a right-inverse function of the complex exponential:
-<math display="block">\int_0^{t_0}|\gamma'(t)| \, dt = \int_0^{t_0} |i\exp(it)| \, dt = t_0,</math>
+<math display="block">e^{\log z} =z. </math>
-starting from {{math|1=''z'' = 1}} in the complex plane and going counterclockwise.  Based on these observations and the fact that the measure of an angle in radians is the arc length on the unit circle subtended by the angle, it is easy to see that, restricted to real arguments, the sine and cosine functions as defined above coincide with the sine and cosine functions as introduced in elementary mathematics via geometric notions.
+However, since the complex logarithm is a [[Multivalued function|multivalued function]], one has
+<math display="block">\log e^z= \{z+2ik\pi\mid k\in \Z\},</math>
+and it is difficult to define the complex exponential from the complex logarithm. On the opposite, this is the complex logarithm that is often defined from the complex exponential.
-The complex exponential function is periodic with period {{math|2''πi''}} and <math>\exp(z+2\pi i k)=\exp z</math> holds for all <math>z \in \mathbb{C}, k \in \mathbb{Z}</math>.
+The complex exponential has the following properties:
+<math display="block">\frac 1{e^z}=e^{-z} </math>
+and
+<math display="block">e^z\neq 0\quad \text{for every } z\in \C .</math>
+It is [[Periodic function|periodic function]] of period {{tmath|2i\pi}}; that is
+<math display="block">e^{z+2ik\pi} =e^z \quad \text{for every } k\in \Z.</math>
+This results from [[Euler's identity]] {{tmath|1=e^{i\pi}=-1}} and the functional identity.
-When its domain is extended from the real line to the complex plane, the exponential function retains the following properties:
+The [[Complex conjugate|complex conjugate]] of the complex exponential is
-<math display="block">\begin{align}
+<math display="block">\overline{e^z}=e^{\overline z}.</math>
-   & e^{z + w} = e^z e^w\, \\[5pt]
+Its modulus is
-   & e^0 = 1\, \\[5pt]
+<math display="block">|e^z|= e^{\Re (z)},</math>
-   & e^z \ne 0 \\[5pt]
+where {{tmath|\Re(z)}} denotes the real part of {{tmath|z}}.
-   & \frac{d}{dz} e^z = e^z \\[5pt]
-   & \left(e^z\right)^n = e^{nz}, n \in \mathbb{Z}
- \end{align} </math>
-for all <math display=inline> w,z\in\mathbb C.</math>
+===Relationship with trigonometry===
+Complex exponential and [[Trigonometric function|trigonometric function]]s are strongly related by [[Euler's formula]]:
+<math display="block">e^{it} =\cos(t)+i\sin(t). </math>
-Extending the natural logarithm to complex arguments yields the [[Complex logarithm|complex logarithm]] {{math|log ''z''}}, which is a [[Multivalued function|multivalued function]].
+This formula provides the decomposition of complex exponentials into real and imaginary parts:
+<math display="block">e^{x+iy} = e^{x}e^{iy} = e^x\,\cos y + i e^x\,\sin y.</math>
-We can then define a more general exponentiation:
+The trigonometric functions can be expressed in terms of complex exponentials:
-<math display="block">z^w = e^{w \log z}</math>
+<math display="block">\begin{align}
-for all complex numbers {{math|''z''}} and {{math|''w''}}. This is also a multivalued function, even when {{math|''z''}} is real. This distinction is problematic, as the multivalued functions {{math|log ''z''}} and {{math|''z''<sup>''w''</sup>}} are easily confused with their single-valued equivalents when substituting a real number for {{math|''z''}}. The rule about multiplying exponents for the case of positive real numbers must be modified in a multivalued context:
+\cos x &= \frac{e^{ix}+e^{-ix}}2\\
-{{block indent|em=1.5|text={{math|(''e{{sup|z}}''){{su|p=''w''}} ≠ ''e{{sup|zw}}''}}, but rather {{math|1=(''e{{sup|z}}''){{su|p=''w''}} = ''e''{{sup|(''z'' + 2''niπ'')''w''}}}} multivalued over integers {{math|''n''}}}}
+\sin x &= \frac{e^{ix}-e^{-ix}}{2i}\\
+\tan x &= i\,\frac{1-e^{2ix}}{1+e^{2ix}}
+\end{align}</math>
-See [[Exponentiation#Failure of power and logarithm identities|failure of power and logarithm identities]] for more about problems with combining powers.
+In these formulas, {{tmath|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.<ref name="Apostol_1974"/>
-The exponential function maps any line in the complex plane to a [[Logarithmic spiral|logarithmic spiral]] in the complex plane with the center at the [[Origin (mathematics)|origin]]. Two special cases exist: when the original line is parallel to the real axis, the resulting spiral never closes in on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius.
+===Plots===
 <gallery caption="3D plots of real part, imaginary part, and modulus of the exponential function" class="center" mode="packed" style="text-align:left" heights="150px">
-  Image:ExponentialAbs_real_SVG.svg| {{math|1=''z'' = Re(''e''<sup>''x'' + ''iy''</sup>)}}
+  Image:ExponentialAbs_real_SVG.svg| {{math|1=''z'' = Re(''e''{{isup|''x'' + ''iy''}})}}
-  Image:ExponentialAbs_image_SVG.svg| {{math|1=''z'' = Im(''e''<sup>''x'' + ''iy''</sup>)}}
+  Image:ExponentialAbs_image_SVG.svg| {{math|1=''z'' = Im(''e''{{isup|''x'' + ''iy''}})}}
-  Image:ExponentialAbs_SVG.svg| {{math|1=''z'' = {{abs|''e''<sup>''x'' + ''iy''</sup>}}}}
+  Image:ExponentialAbs_SVG.svg| {{math|1=''z'' = {{abs|''e''{{isup|''x'' + ''iy''}}}}}}
 </gallery>
@@ Line 216: / Line 236: @@
 Starting with a color-coded portion of the <math>xy</math> domain, the following are depictions of the graph as variously projected into two or three dimensions.
-<gallery caption="Graphs of the complex exponential function" class="center" mode="packed" style="text-align:left" heights="200px">
+<gallery class="center" mode="packed" style="text-align:left" heights="200px" caption="Graphs of the complex exponential function">
-File: Complex exponential function graph domain xy dimensions.svg|Checker board key:<br> <math>x> 0:\; \text{green}</math><br> <math>x< 0:\; \text{red}</math><br><math> y> 0:\; \text{yellow}</math><br><math> y< 0:\; \text{blue}</math>
+File: Complex exponential function graph domain xy dimensions.svg|Checker board key:<br> <math>x> 0:\; \text{green}</math><br> <math>x< 0:\; \text{red}</math><br><math>y> 0:\; \text{yellow}</math><br><math>y< 0:\; \text{blue}</math>
 File: Complex exponential function graph range vw dimensions.svg|Projection onto the range complex plane (V/W). Compare to the next, perspective picture.
-File: Complex exponential function graph horn shape xvw dimensions.jpg|Projection into the <math>x</math>, <math>v</math>, and <math>w</math> dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image).
+File: Complex exponential function graph horn shape xvw dimensions.jpg|Projection into the <math>x</math>, <math>v</math>, and <math>w</math> dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image)|alt=Projection into the                 <nowiki> </nowiki>       x             <nowiki> </nowiki>   {\displaystyle x}    ,                 <nowiki> </nowiki>       v             <nowiki> </nowiki>   {\displaystyle v}    , and                 <nowiki> </nowiki>       w             <nowiki> </nowiki>   {\displaystyle w}    <nowiki> </nowiki>dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image).
-File: Complex exponential function graph spiral shape yvw dimensions.jpg|Projection into the <math>y</math>, <math>v</math>, and <math>w</math> dimensions, producing a spiral shape. (<math>y</math> range extended to ±2{{pi}}, again as  2-D perspective image).
+File: Complex exponential function graph spiral shape yvw dimensions.jpg|Projection into the <math>y</math>, <math>v</math>, and <math>w</math> dimensions, producing a spiral shape (<math>y</math> range extended to ±2{{pi}}, again as 2-D perspective image)|alt=Projection into the                 <nowiki> </nowiki>       y             <nowiki> </nowiki>   {\displaystyle y}    ,                 <nowiki> </nowiki>       v             <nowiki> </nowiki>   {\displaystyle v}    , and                 <nowiki> </nowiki>       w             <nowiki> </nowiki>   {\displaystyle w}    <nowiki> </nowiki>dimensions, producing a spiral shape. (                <nowiki> </nowiki>       y             <nowiki> </nowiki>   {\displaystyle y}    <nowiki> </nowiki>range extended to ±2π, again as 2-D perspective image).
 </gallery>
@@ Line 237: / Line 257: @@
 The fourth image shows the graph extended along the imaginary <math>y</math> axis.  It shows that the graph's surface for positive and negative <math>y</math> values doesn't really meet along the negative real <math>v</math> axis, but instead forms a spiral surface about the <math>y</math> axis.  Because its <math>y</math> values have been extended to {{math|±2''π''}}, this image also better depicts the 2π periodicity in the imaginary <math>y</math> value.
-===Computation of {{math|''a''<sup>''b''</sup>}} where both {{math|''a''}} and {{math|''b''}} are complex===
-{{Main|Exponentiation}}
-Complex exponentiation {{math|''a''<sup>''b''</sup>}} can be defined by converting {{math|''a''}} to polar coordinates and using the identity {{math|1=(''e''<sup>ln ''a''</sup>){{su|p=''b''}} = ''a''<sup>''b''</sup>}}:
-<math display="block">a^b = \left(re^{\theta i}\right)^b = \left(e^{(\ln r) + \theta i}\right)^b = e^{\left((\ln r) + \theta i\right)b}</math>
-However, when {{math|''b''}} is not an integer, this function is [[Multivalued function|multivalued]], because {{math|''θ''}} is not unique (see ''{{slink|Exponentiation#Failure of power and logarithm identities}}'').
 ==Matrices and Banach algebras==
-The power series definition of the exponential function makes sense for square [[Matrix (mathematics)|matrices]] (for which the function is called the [[Matrix exponential|matrix exponential]]) and more generally in any unital [[Banach algebra]] {{math|''B''}}.  In this setting, {{math|1=''e''<sup>0</sup> = 1}}, and {{math|''e''<sup>''x''</sup>}} is invertible with inverse {{math|''e''<sup>−''x''</sup>}} for any {{math|''x''}} in {{math|''B''}}.  If {{math|1=''xy'' = ''yx''}}, then {{math|1=''e''<sup>''x'' + ''y''</sup> = ''e''<sup>''x''</sup>''e''<sup>''y''</sup>}}, but this identity can fail for noncommuting {{math|''x''}} and {{math|''y''}}.
+The power series definition of the exponential function makes sense for square [[Matrix (mathematics)|matrices]] (for which the function is called the [[Matrix exponential|matrix exponential]]) and more generally in any unital [[Banach algebra]] {{math|''B''}}.  In this setting, {{math|1=''e''{{isup|0}} = 1}}, and {{math|''e''{{isup|''x''}}}} is invertible with inverse {{math|''e''{{isup|−''x''}}}} for any {{math|''x''}} in {{math|''B''}}.  If {{math|1=''xy'' = ''yx''}}, then {{math|1=''e''{{isup|''x'' + ''y''}} = ''e''{{isup|''x''}}''e''{{isup|''y''}}}}, but this identity can fail for noncommuting {{math|''x''}} and {{math|''y''}}.
-Some alternative definitions lead to the same function.  For instance, {{math|''e''<sup>''x''</sup>}} can be defined as
+Some alternative definitions lead to the same function.  For instance, {{math|''e''{{isup|''x''}}}} can be defined as
 <math display="block">\lim_{n \to \infty} \left(1 + \frac{x}{n} \right)^n .</math>
-Or {{math|''e''<sup>''x''</sup>}} can be defined as {{math|''f''<sub>''x''</sub>(1)}}, where {{math|''f''<sub>''x''</sub> : '''R''' → ''B''}} is the solution to the differential equation {{math|1={{sfrac|''df<sub>x</sub>''|''dt''}}(''t'') = ''x{{space|hair}}f''<sub>''x''</sub>(''t'')}}, with initial condition {{math|1=''f''<sub>''x''</sub>(0) = 1}}; it follows that {{math|1=''f''<sub>''x''</sub>(''t'') = ''e''<sup>''tx''</sup>}} for every {{mvar|t}} in {{math|'''R'''}}.
+Or {{math|''e''{{isup|''x''}}}} can be defined as {{math|''f''<sub>''x''</sub>(1)}}, where {{math|''f''<sub>''x''</sub> : '''R''' → ''B''}} is the solution to the differential equation {{math|1={{sfrac|''df''<sub>''x''</sub>|''dt''}}(''t'') = ''x{{space|hair}}f''<sub>''x''</sub>(''t'')}}, with initial condition {{math|1=''f''<sub>''x''</sub>(0) = 1}}; it follows that {{math|1=''f''<sub>''x''</sub>(''t'') = ''e''{{isup|''tx''}}}} for every {{mvar|t}} in {{math|'''R'''}}.
 ==Lie algebras==
@@ Line 259: / Line 272: @@
 ==Transcendency==
-The function {{math|''e''<sup>''z''</sup>}} is not in the rational function ring <math>\C(z)</math>: it is not the quotient of two polynomials with complex coefficients.
+The function {{math|''e''{{isup|''z''}}}} is a [[Transcendental function|transcendental function]], which means that it is not a root of a polynomial over the [[Ring (mathematics)|ring]] of the rational fractions <math>\C(z).</math>
-If {{math|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub><nowiki/>}} are distinct complex numbers, then {{math|''e''<sup>''a''<sub>1</sub>''z''</sup>, ..., ''e''<sup>''a''<sub>''n''</sub>''z''</sup><nowiki/>}} are linearly independent over <math>\C(z)</math>, and hence {{math|''e''<sup>''z''</sup>}} is [[Transcendental function|transcendental]] over <math>\C(z)</math>.
+If {{math|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub><nowiki/>}} are distinct complex numbers, then {{math|''e''<sup>''a''<sub>1</sub>''z''</sup>, ..., ''e''<sup>''a''<sub>''n''</sub>''z''</sup><nowiki/>}} are linearly independent over <math>\C(z)</math>, and hence {{math|''e''{{isup|''z''}}}} is [[Transcendental function|transcendental]] over <math>\C(z)</math>.
 =={{anchor|exp|expm1}}Computation==
-When computing (an approximation of) the exponential function near the argument {{math|0}}, the result will be close to 1, and computing the value of the difference <math>e^x-1</math> with [[Floating-point arithmetic|floating-point arithmetic]] may lead to the loss of (possibly all) [[Significant figures|significant figures]], producing a large calculation error, possibly even a meaningless result.
+The Taylor series definition above is generally efficient for computing (an approximation of) <math>e^x</math>. However, when computing near the argument <math>x=0</math>, the result will be close to 1, and computing the value of the difference <math>e^x-1</math> with [[Floating-point arithmetic|floating-point arithmetic]] may lead to the loss of (possibly all) [[Significant figures|significant figures]], producing a large relative error, possibly even a meaningless result.
-Following a proposal by [[Biography:William Kahan|William Kahan]], it may thus be useful to have a dedicated routine, often called <code>expm1</code>, for computing  {{math|''e<sup>x</sup>'' − 1}} directly, bypassing computation of {{math|''e''<sup>''x''</sup>}}. For example, if the exponential is computed by using its [[Taylor series]]
+Following a proposal by [[Biography:William Kahan|William Kahan]], it may thus be useful to have a dedicated routine, often called <code>expm1</code>, which computes  {{math|''e<sup>x</sup>'' − 1}} directly, bypassing computation of {{math|''e''{{isup|''x''}}}}. For example,
-<math display="block">e^x = 1 + x + \frac {x^2}2 + \frac{x^3}6 + \cdots + \frac{x^n}{n!} + \cdots,</math>
+one may use the Taylor series:
-one may use the Taylor series of <math>e^x-1</math>:
 <math display="block">e^x-1=x+\frac {x^2}2 + \frac{x^3}6+\cdots +\frac{x^n}{n!}+\cdots.</math>
@@ Line 275: / Line 287: @@
 In addition to base {{math|''e''}}, the [[IEEE 754-2008]] standard defines similar exponential functions near 0 for base 2 and 10: <math>2^x - 1</math> and <math>10^x - 1</math>.
-A similar approach has been used for the logarithm (see lnp1).<ref group="nb" name="Alternative_funcs"/>
+A similar approach has been used for the logarithm; see log1p.
 An identity in terms of the hyperbolic tangent,
 <math display="block">\operatorname{expm1} (x) = e^x - 1 = \frac{2 \tanh(x/2)}{1 - \tanh(x/2)},</math>
 gives a high-precision value for small values of {{math|''x''}} on systems that do not implement {{math|expm1(''x'')}}.
+===Continued fractions===
+The exponential function can also be computed with [[Continued fraction|continued fraction]]s.
+A continued fraction for {{math|''e''{{isup|''x''}}}} can be obtained via [[Euler's continued fraction formula|an identity of Euler]]:
+<math display="block"> e^x = 1 + \cfrac{x}{1 - \cfrac{x}{x + 2 - \cfrac{2x}{x + 3 - \cfrac{3x}{x + 4 - \ddots}}}}</math>
+The following [[Generalized continued fraction|generalized continued fraction]] for {{math|''e''{{isup|''z''}}}} also received by Euler
+<ref>A. N. Khovanski, The applications of continued fractions and their Generalisation to problemes in approximation theory,1963, Noordhoff, Groningen, The Netherlands</ref>
+converges more quickly:<ref name="Lorentzen_2008"/>
+<math display="block"> e^z = 1 + \cfrac{2z}{2 - z + \cfrac{z^2}{6 + \cfrac{z^2}{10 + \cfrac{z^2}{14 + \ddots}}}}</math>
+or, by applying the substitution {{math|1=''z'' = {{sfrac|''x''|''y''}}}}:
+<math display="block">  e^\frac{x}{y} = 1 + \cfrac{2x}{2y - x + \cfrac{x^2} {6y + \cfrac{x^2} {10y + \cfrac{x^2} {14y + \ddots}}}}</math>
+with a special case for {{math|1=''z'' = 2}}:
+<math display="block">  e^2 = 1 + \cfrac{4}{0 + \cfrac{2^2}{6 + \cfrac{2^2}{10 + \cfrac{2^2}{14 + \ddots }}}} = 7 + \cfrac{2}{5 + \cfrac{1}{7 + \cfrac{1}{9 + \cfrac{1}{11 + \ddots }}}}</math>
+This formula also converges, though more slowly, for {{math|''z'' > 2}}. For example:
+<math display="block"> e^3 = 1 + \cfrac{6}{-1 + \cfrac{3^2}{6 + \cfrac{3^2}{10 + \cfrac{3^2}{14 + \ddots }}}} = 13 + \cfrac{54}{7 + \cfrac{9}{14 + \cfrac{9}{18 + \cfrac{9}{22 + \ddots }}}}</math>
 ==See also==
@@ Line 291: / Line 322: @@
 * [[List of integrals of exponential functions]]
 * [[Mittag-Leffler function]], a generalization of the exponential function
-* [[P-adic exponential function|{{math|''p''}}-adic exponential function]]
+* [[p-adic exponential function|{{math|''p''}}-adic exponential function]]
-* Padé table for exponential function – Padé approximation of exponential function by a fraction of polynomial functions
+* Padé table for exponential function – [[Padé approximation]] of exponential function by a fraction of polynomial functions
+* [[Phase factor]]
 {{div col end}}
 ==Notes==
-{{Reflist|group="nb"|refs=
+{{Notelist}}
-<ref group="nb" name="Alternative_funcs">A similar approach to reduce [[Round-off error|round-off error]]s of calculations for certain input values of trigonometric functions consists of using the less common trigonometric functions [[Versine|versine]], [[Vercosine|vercosine]], [[Coversine|coversine]], [[Covercosine|covercosine]], [[Haversine|haversine]], [[Havercosine|havercosine]], [[Hacoversine|hacoversine]], [[Hacovercosine|hacovercosine]], [[Exsecant|exsecant]] and [[Excosecant|excosecant]].</ref>
-<ref group="nb" name="Logx">In pure mathematics, the notation {{math|log ''x''}} generally refers to the natural logarithm of {{mvar|x}} or a logarithm in general if the base is immaterial.</ref>
-<ref group="nb" name="lnx">The notation {{math|ln ''x''}} is the ISO standard and is prevalent in the natural sciences and secondary education (US).  However, some mathematicians (for example, [[Biography:Paul Halmos|Paul Halmos]]) have criticized this notation and prefer to use {{math|log ''x''}} for the natural logarithm of {{mvar|x}}.</ref>
-}}
 ==References==
-{{Reflist|refs=
+{{reflist|refs=
-<ref name="Goldstein_2006">{{cite book |author-last1=Goldstein |author-first1=Larry Joel |author-last2=Lay |author-first2=David C. |author-last3=Schneider |author-first3=David I. |author-last4=Asmar |author-first4=Nakhle H. |title=Brief calculus and its applications |edition=11th |publisher=Prentice–Hall |date=2006 |isbn=978-0-13-191965-5 |url=https://books.google.com/books?id=F82cPAAACAAJ&pg=PA448}} (467 pages)</ref>
+<!-- <ref name="Rudin_1976">{{Cite book |title=Principles of Mathematical Analysis |author-last=Rudin |author-first=Walter |publisher=McGraw-Hill |date=1976 |isbn=978-0-07-054235-8 |location=New York |pages=182 |url=https://archive.org/details/PrinciplesOfMathematicalAnalysis}}</ref> -->
-<ref name="Courant_1996">{{cite book |quote=''This natural exponential function is identical with its derivative.'' This is really the source of all the properties of the exponential function, and the basic reason for its importance in applications… |page=448 |author-last1=Courant |author-last2=Robbins |title=What is Mathematics? An Elementary Approach to Ideas and Methods |editor-last=Stewart |edition=2nd revised |publisher=[[Organization:Oxford University Press|Oxford University Press]] |date=1996 |isbn=978-0-13-191965-5}}</ref>
-<ref name="Durell_1911">{{cite book |author-last1=Converse |author-first1=Henry Augustus |author-last2=Durell |author-first2=Fletcher |title=Plane and Spherical Trigonometry |publisher=C. E. Merrill Company |date=1911 |page=[https://archive.org/details/planeandspheric00duregoog/page/n18 12] |series=Durell's mathematical series |url=https://archive.org/details/planeandspheric00duregoog |quote=Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it, (called its antilogarithm) ...}} [https://books.google.com/books/about/Plane_and_spherical_trigonometry.html?id=9NwOAAAAYAAJ]</ref>
-<ref name="Rudin_1976">{{Cite book |title=Principles of Mathematical Analysis |author-last=Rudin |author-first=Walter |publisher=McGraw-Hill |date=1976 |isbn=978-0-07-054235-8 |location=New York |pages=182 |url=https://archive.org/details/PrinciplesOfMathematicalAnalysis}}</ref>
 <ref name="Rudin_1987">{{cite book |title=Real and complex analysis |author-last=Rudin |author-first=Walter |date=1987 |publisher=McGraw-Hill |isbn=978-0-07-054234-1 |edition=3rd |location=New York |page=1 |url=https://archive.org/details/RudinW.RealAndComplexAnalysis3e1987}}</ref>
 <ref name="Maor">{{cite book |author-first=Eli |author-last=Maor |title=e: the Story of a Number |page=156}}</ref>
-<ref name="O'Connor_2001">{{cite web |title=The number e |author-first1=John J. |author-last1=O'Connor |author-first2=Edmund F. |author-last2=Robertson |publisher=University of St Andrews, Scotland |work=School of Mathematics and Statistics  |date=September 2001 |url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/e.html |access-date=2011-06-13}}</ref>
+<ref name="O'Connor_2001">{{MacTutor|class=HistTopics|id=e}}
+</ref>
 <ref name="Lorentzen_2008">{{cite book |title=Continued Fractions |chapter=A.2.2 The exponential function. |author-first1=L. |author-last1=Lorentzen|author-first2=H. |author-last2=Waadeland |series=Atlantis Studies in Mathematics |date=2008 |volume=1 |doi=10.2991/978-94-91216-37-4 |page=268 |isbn=978-94-91216-37-4 |chapter-url=https://link.springer.com/content/pdf/bbm%3A978-94-91216-37-4%2F1}}</ref>
 <ref name="Apostol_1974">{{Cite book |title=Mathematical Analysis |url=https://archive.org/details/mathematicalanal00apos_530 |url-access=limited |author-last=Apostol |author-first=Tom M. |publisher=Addison Wesley |date=1974 |isbn=978-0-201-00288-1 |edition=2nd |location=Reading, Mass. |pages=[https://archive.org/details/mathematicalanal00apos_530/page/n32 19]}}</ref>

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