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{{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}}
{{Use dmy dates|date=August 2019|cs1-dates=y}}
{{Infobox mathematical function
| name = Exponential
| image = Image:exp.svg
| imagealt = Graph of the exponential function
| caption = Graph of the exponential function
| general_definition = <math>\exp z = e^{z}</math>
| motivation_of_creation =
| fields_of_application =
| ___domain = <math>\mathbb{C}</math>
| range = <math>\begin{cases} (0,\infty) & \text{for }z \in \mathbb{R} \\ \mathbb{C} \setminus \{0\} & \text{for }z \in \mathbb{C} \end{cases}</math>
| zero = 1
| vr1 = 1
| 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>
| antiderivative = <math>\int \exp z\,dz = \exp z + C</math>
| taylor_series = <math>\exp z = \sum_{n=0}^\infty\frac{z^n}{n!}</math>
}}
In [[mathematics]], the '''exponential function''' is the unique [[real function]] which maps [[0|zero]] to [[1|one]] and has a [[derivative (mathematics)|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 (mathematics)|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 converts sums to products: it maps the [[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]], the [[natural logarithm]], {{tmath|\ln}} or {{tmath|\log}}, converts products to sums: {{tmath|1= \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 {{tmath|1=f(x) = b^x}}, which is [[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)}}.
The exponential function can be generalized to accept [[complex number]]s as arguments. This reveals relations between multiplication of complex numbers, rotations in the [[complex plane]], and [[trigonometry]]. [[Euler's formula]] {{tmath|1= \exp i\theta = \cos\theta + i\sin\theta}} expresses and summarizes these relations.
The exponential function can be even further generalized to accept other types of arguments, such as [[Matrix exponentiation|matrices]] and elements of [[Exponential map (Lie theory)|Lie algebras]].
==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]]. The equation <math>\tfrac{d}{dx}e^x = e^x</math> means that the [[slope]] of the [[tangent]] to the graph at each point is equal to its height (its {{mvar|y}}-coordinate) at that point.
==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.
===Differential equation===
[[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]] of the tangent, this implies that all green [[right triangle]]s have a base length of 1.]]
One of the simplest definitions is: The ''exponential function'' is the ''unique'' [[differentiable function]] that equals its [[derivative]], and takes the value {{math|1}} for the value {{math|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 {{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]]. It follows that {{tmath|f/g}} is constant; this constant is {{math|1}} since {{tmath|1=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
:<math>\begin{align}
\ln (\exp x)&=x\\
\exp(\ln y)&=y
\end{align}</math>
for every [[real number]] <math>x</math> and every positive real number <math>y.</math>
===Power series===
''The exponential function is the sum of the [[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>
\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]] 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]]. 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]].
===Functional equation===
''The exponential satisfies the [[functional equation]]:''
<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.
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}}.
===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]].
===Properties===
''[[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
<math display=block>\frac 1{e^x}=e^{-x}.</math>
''Positiveness:'' {{tmath|e^x>0}} for every real number {{tmath|x}}. This results from the [[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]].
''Extension of [[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>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.
==General exponential functions==
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]] 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 {{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}}.
These most general exponential functions are the [[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>
[[Image:Exponenciala priklad.png|thumb|200px|right|Exponential functions with bases 2 and 1/2]]
The ''base'' of an exponential function is the ''base'' of the [[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.
===In applications===
The last characterization is important in [[empirical science]]s, as allowing a direct [[experimental]] test whether a function is an exponential function.
Exponential [[exponential growth|growth]] or [[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 [[Malthusian catastrophe]], [[compound interest#Continuous compounding|continuously compounded interest]], and [[radioactive decay]].
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" />
===Equivalence proof===
For proving the equivalence of the above properties, one can proceed as follows.
The two first characterizations are equivalent, since, if {{tmath|1=b=e^k}} and {{tmath|1= k=\ln b}}, one has
<math display=block>e^{kx}= (e^k)^x= b^x.</math>
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 {{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]] 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>
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{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}}.
==Compound interest==
The earliest occurrence of the exponential function was in [[Jacob Bernoulli]]'s study of [[compound interest]]s in 1683.<ref name="O'Connor_2001"/>
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}}.
The exponential function is involved as follows in the computation of [[compound interest#Continuous compounding|continuously compounded interests]].
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">\exp x = \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^{n}</math>
first given by [[Leonhard Euler]].<ref name="Maor"/>
==Differential equations==
{{main|Linear differential equation}}
Exponential functions occur very often in solutions of [[differential equation]]s.
The exponential functions can be defined as solutions of [[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.
The solutions of an equation of the form
<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]] 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==
{{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>.]]
The exponential function can be naturally extended to a [[complex function]], which is a function with the [[complex number]]s as [[___domain of a function|___domain]] and [[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'''.
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 {{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 ''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]].
The complex exponential function is the [[limit (mathematics)|limit]]
<math display="block">e^z = \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n</math>
As with the real exponential function (see {{slink||Functional equation}} above), the complex exponential satisfies the functional equation
<math display=block>\exp(z+w)= \exp(z)\cdot \exp(w).</math>
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>
The [[complex logarithm]] is a [[left inverse function|right-inverse function ]] of the complex exponential:
<math display="block">e^{\log z} =z. </math>
However, since the complex logarithm is a [[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 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.
The [[complex conjugate]] of the complex exponential is
<math display="block">\overline{e^z}=e^{\overline z}.</math>
Its modulus is
<math display="block">|e^z|= e^{\Re (z)},</math>
where {{tmath|\Re(z)}} denotes the real part of {{tmath|z}}.
===Relationship with trigonometry===
Complex exponential and [[trigonometric function]]s are strongly related by [[Euler's formula]]:
<math display="block">e^{it} =\cos(t)+i\sin(t). </math>
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>
The trigonometric functions can be expressed in terms of complex exponentials:
<math display="block">\begin{align}
\cos x &= \frac{e^{ix}+e^{-ix}}2\\
\sin x &= \frac{e^{ix}-e^{-ix}}{2i}\\
\tan x &= i\,\frac{1-e^{2ix}}{1+e^{2ix}}
\end{align}</math>
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"/>
===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''{{isup|''x'' + ''iy''}})}}
Image:ExponentialAbs_image_SVG.svg| {{math|1=''z'' = Im(''e''{{isup|''x'' + ''iy''}})}}
Image:ExponentialAbs_SVG.svg| {{math|1=''z'' = {{abs|''e''{{isup|''x'' + ''iy''}}}}}}
</gallery>
Considering the complex exponential function as a function involving four real variables:
<math display="block">v + i w = \exp(x + i y)</math>
the graph of the exponential function is a two-dimensional surface curving through four dimensions.
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 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 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)|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)|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>
The second image shows how the ___domain complex plane is mapped into the range complex plane:
* zero is mapped to 1
* the real <math>x</math> axis is mapped to the positive real <math>v</math> axis
* the imaginary <math>y</math> 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 <math>x</math> axis. It shows the graph is a surface of revolution about the <math>x</math> 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 <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.
==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]]) 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''{{isup|''x''}}}} can be defined as
<math display="block">\lim_{n \to \infty} \left(1 + \frac{x}{n} \right)^n .</math>
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==
Given a [[Lie group]] {{math|''G''}} and its associated [[Lie algebra]] <math>\mathfrak{g}</math>, the [[exponential map (Lie theory)|exponential map]] is a map <math>\mathfrak{g}</math> {{math|↦ ''G''}} satisfying similar properties. In fact, since {{math|'''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 {{math|GL(''n'','''R''')}} of invertible {{math|''n'' × ''n''}} matrices has as Lie algebra {{math|M(''n'','''R''')}}, the space of all {{math|''n'' × ''n''}} matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.
The identity <math>\exp(x+y)=\exp(x)\exp(y)</math> can fail for Lie algebra elements {{math|''x''}} and {{math|''y''}} that do not commute; the [[Baker–Campbell–Hausdorff formula]] supplies the necessary correction terms.
==Transcendency==
The function {{math|''e''{{isup|''z''}}}} is a [[transcendental function]], which means that it is not a [[polynomial root|root]] of a polynomial over the [[ring (mathematics)|ring]] of the [[rational fraction]]s <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==
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]] 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 <code>expm1</code>, which computes {{math|''e<sup>x</sup>'' − 1}} directly, bypassing computation of {{math|''e''{{isup|''x''}}}}. For example,
one may use the Taylor series:
<math display="block">e^x-1=x+\frac {x^2}2 + \frac{x^3}6+\cdots +\frac{x^n}{n!}+\cdots.</math>
This was first implemented in 1979 in the [[Hewlett-Packard]] [[HP-41C]] calculator, and provided by several calculators,<ref name="HP48_AUR"/><ref name="HP50_AUR"/> [[operating system]]s (for example [[Berkeley UNIX 4.3BSD]]<ref name="Beebe_2017"/>), [[computer algebra system]]s, and programming languages (for example [[C99]]).<ref name="Beebe_2002"/>
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 [[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]]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]] for {{math|''e''{{isup|''z''}}}} 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==
{{Portal|Mathematics}}
{{div col}}
* [[Carlitz exponential]], a characteristic {{math|''p''}} analogue
* {{annotated link|Double exponential function}}
* {{annotated link|Exponential field}}
* [[Gaussian function]]
* [[Half-exponential function]], a compositional square root of an exponential function
* {{annotated link|Lambert W function#Solving equations}} - Used for solving exponential equations
* [[List of exponential topics]]
* [[List of integrals of exponential functions]]
* [[Mittag-Leffler function]], a generalization of the 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
* [[Phase factor]]
{{div col end}}
==Notes==
{{Notelist}}
==References==
{{reflist|refs=
<!-- <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 |author-link=Eli Maor |title=e: the Story of a Number |page=156}}</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|author1-link= Lisa 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>
<ref name="Beebe_2002">{{cite web |title=Computation of expm1 = exp(x)−1 |author-first=Nelson H. F. |author-last=Beebe |publisher=Department of Mathematics, Center for Scientific Computing, University of Utah |___location=Salt Lake City, Utah, USA |date=2002-07-09 |version=1.00 |url=http://www.math.utah.edu/~beebe/reports/expm1.pdf |access-date=2015-11-02}}</ref>
<ref name="Beebe_2017">{{cite book |author-first=Nelson H. F. |author-last=Beebe |title=The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library |chapter=Chapter 10.2. Exponential near zero |date=2017-08-22 |___location=Salt Lake City, UT, USA |publisher=[[Springer International Publishing AG]] |edition=1 |lccn=2017947446 |isbn=978-3-319-64109-6 |doi=10.1007/978-3-319-64110-2 |pages=273–282 |s2cid=30244721 |quote=Berkeley UNIX 4.3BSD introduced the expm1() function in 1987.}}</ref>
<ref name="HP48_AUR">{{cite book |title=HP 48G Series – Advanced User's Reference Manual (AUR) |publisher=[[Hewlett-Packard]] |edition=4 |date=December 1994 |id=HP 00048-90136, 0-88698-01574-2 |orig-year=1993<!-- edition 1 (1993-07) --> |url=http://www.hpcalc.org/details.php?id=6036 |access-date=2015-09-06}}</ref>
<ref name="HP50_AUR">{{cite book |title=HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR) |publisher=[[Hewlett-Packard]] |edition=2 |date=2009-07-14 |orig-year=2005<!-- first published: Edition 1 (2005–09) --> |id=HP F2228-90010 |url=http://www.hpcalc.org/details.php?id=7141 |access-date=2015-10-10}} [http://holyjoe.net/hp/HP_50g_AUR_v2_English_searchable.pdf]</ref>
<ref name="McGrawHill_2007">{{cite encyclopedia |title=McGraw-Hill Encyclopedia of Science & Technology |edition=10th |___location=New York |publisher=[[McGraw-Hill]] |date=2007 |isbn=978-0-07-144143-8}}</ref>
<ref name="Serway-Moses-Moyer_1989">{{cite book |first1=Raymond A. |last1=Serway |first2=Clement J. |last2=Moses |first3=Curt A. |last3=Moyer |date=1989 |isbn=0-03-004844-3 |title=Modern Physics |publisher=[[Harcourt Brace Jovanovich]] |___location=Fort Worth |page=384}}</ref>
<ref name="Simmons_1972">{{cite book |first1=George F. |last1=Simmons |author-link=George F. Simmons |date=1972 |title=Differential Equations with Applications and Historical Notes |publisher=[[McGraw-Hill]] |___location=New York |lccn=75173716 |page=15}}</ref>
}}
==External links==
* {{springer|title=Exponential function|id=p/e036910}}
{{Calculus topics}}
{{Authority control}}
[[Category:Elementary special functions]]
[[Category:Analytic functions]]
[[Category:Exponentials]]
[[Category:Special hypergeometric functions]]
[[Category:E (mathematical constant)]]
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