Exponential function: Difference between revisions
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{{Infobox mathematical function | {{Infobox mathematical function | ||
| name = Exponential | | name = Exponential | ||
| image = Image:exp.svg | | image = Image:exp.svg{{!}}class=skin-invert-image | ||
| imagealt = Graph of the exponential function | | imagealt = Graph of the exponential function | ||
| caption = Graph of the exponential function | | caption = Graph of the exponential function | ||
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| reciprocal = <math>\exp(-z)</math> | | reciprocal = <math>\exp(-z)</math> | ||
| inverse = [[Natural logarithm]], [[Complex logarithm]] | | inverse = [[Natural logarithm]], [[Complex logarithm]] | ||
| derivative = <math>\ | | derivative = <math>\frac{\mathrm{d}}{\mathrm{d}\!\,z} \exp z = \exp z</math> | ||
| antiderivative = <math>\int \exp z\,dz = \exp z + C</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> | | 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 [[ | In [[mathematics]], the '''exponential function''' is the unique [[real function]] which maps [[0|zero]] to [[1|one]] and has a [[derivative]] everywhere equal to its value. It is denoted {{tmath|e^x}} or {{tmath|\exp x}}; the latter is preferred when the argument {{tmath|x}} is a complicated expression.<ref>{{cite web |title=Reviews of Modern Physics Style Guide |url=https://cdn.journals.aps.org/files/rmpguide.pdf |publisher=American Physical Society |access-date=30 December 2025 |location=XVI.B.1(d) |page=18 |quote=Which form to use, {{tmath|e}} or {{tmath|\exp}}, is determined by the number of characters and the complexity of the argument. The {{tmath|e}} form is appropriate when the argument is short and simple, i.e., <math>e^{i \mathbf{k}\cdot\mathbf{r}}</math>, whereas {{tmath|\exp}} should be used if the argument is more complicated.}}</ref><ref>{{cite book |author1=T. W. Chaundy |author2=P. R. Barrett |author3=Charles Batey |title=The Printing of Mathematics |date=1954 |publisher=Oxford University Press |page=31 |url=https://archive.org/details/printingofmathem0000chau/page/31/}}</ref> 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: | The exponential function converts sums to products: {{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 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= | 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= e^{i\theta} = \cos\theta + i\sin\theta}} expresses and summarizes these relations. | ||
The exponential function can be even further generalized to accept other types of arguments, such as [[Matrix exponentiation|matrices]] and elements of [[Exponential map (Lie theory)|Lie algebras]]. | 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]]. | ||
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[[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.]] | [[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.]] | ||
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 | This definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function. | ||
===Inverse of natural logarithm=== | ===Inverse of natural logarithm=== | ||
The exponential function is the [[inverse function]] of the [[natural logarithm]]. That is, | |||
:<math>\begin{align} | :<math>\begin{align} | ||
\ln (\exp x)&=x\\ | \ln (\exp x)&=x\\ | ||
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===Power series=== | ===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> | <math display=block> | ||
\begin{align}\exp(x) &= 1+x+\frac{x^2}{2!}+ \frac{x^3}{3!}+\cdots\\ | \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> | &=\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)]] | [[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> | 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>, by the [[ratio test]]. This shows that the exponential function is defined for every {{tmath|x}}, and is everywhere the sum of its [[Maclaurin series]]. | ||
===Functional equation=== | ===Functional equation=== | ||
The exponential satisfies the [[functional equation]] | |||
<math display=block>\exp(x+y)= \exp(x)\cdot \exp(y) | <math display=block>\exp(x+y)= \exp(x)\cdot \exp(y)</math> | ||
and maps the [[additive identity]] {{math|0}} to the [[multiplicative identity]] {{math|1}}. | |||
<math> f(x)= | The same equation is satisfied by other continuous functions <math>f(x)=b^x</math> that exponentiate their argument with an arbitrary base <math>b</math>.<ref>{{cite book | ||
| last = Jung | first = Soon-Mo | |||
| contribution = Chapter 9: Exponential Functional Equations | |||
| doi = 10.1007/978-1-4419-9637-4_9 | |||
| isbn = 9781441996374 | |||
| series = Springer Optimization and Its Applications | |||
| pages = 207–225 | |||
| publisher = Springer New York | |||
| title = Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis | |||
| year = 2011 | |||
| volume = 48 | |||
}}</ref> Among these functions, the exponential function is characterized by the property that its derivative at {{math|0}} is {{math|1}}.<ref>{{cite book | |||
| last1 = Aczél | first1 = J. | |||
| last2 = Dhombres | first2 = J. | |||
| doi = 10.1017/CBO9781139086578 | |||
| isbn = 0-521-35276-2 | |||
| mr = 1004465 | |||
| page = 10 | |||
| publisher = Cambridge University Press, Cambridge | |||
| series = Encyclopedia of Mathematics and its Applications | |||
| title = Functional Equations in Several Variables | |||
| volume = 31 | |||
| year = 1989}}</ref> | |||
===Limit of integer powers=== | ===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> | <math display=block>\exp(x)=\lim_{n \to +\infty} \left(1+\frac xn\right)^n.</math> | ||
===Properties=== | ===Properties=== | ||
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''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]]. | ''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 | ''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> | <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}}: | 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}}: | ||
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[[Image:Exponenciala priklad.png|thumb|200px|right|Exponential functions with bases 2 and 1/2]] | [[Image:Exponenciala priklad.png|thumb|200px|right|Exponential functions with bases 2 and 1/2]] | ||
The ''base'' of an exponential function | 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=== | ===In applications=== | ||
The last characterization is important in [[empirical science]]s, as allowing a direct [[experimental]] test whether a function is an exponential function. | 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]]. | 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]]. | ||
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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 | 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 | 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> | <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 | 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 | ||
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involve exponential functions in a more sophisticated way, since they have the form | 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> | <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. | 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. | 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== | == Complex exponential == | ||
{{anchor|On the complex plane|Complex plane}} | {{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''}}]] | [[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>.]] | [[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 | 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. | 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 | 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}}: | The ''complex exponential'' is the unique complex function that equals its [[complex derivative]] and takes the value {{tmath|1}} for the argument {{tmath|0}}: | ||
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The ''complex exponential function'' is the sum of the [[series (mathematics)|series]] | 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> | <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 | This series is [[absolutely convergent]] for every complex number {{tmath|z}}. So, the complex exponential is an [[entire function]]. | ||
The complex exponential function is the [[limit (mathematics)|limit]] | 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> | <math display="block">e^z = \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n</math> | ||
As with the real exponential function (see {{section link||Functional equation}} above), the complex exponential satisfies the functional equation | |||
<math display= | <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: | The [[complex logarithm]] is a [[left inverse function|right-inverse function]] of the complex exponential: | ||
<math display="block">e^{\log z} =z. </math> | <math display="block">e^{\log z} =z. </math> | ||
However, since the complex logarithm is a [[multivalued function]], one has | However, since the complex logarithm is a [[multivalued function]], one has | ||
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and | and | ||
<math display="block">e^z\neq 0\quad \text{for every } z\in \C .</math> | <math display="block">e^z\neq 0\quad \text{for every } z\in \C .</math> | ||
It is [[ | It is [[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> | <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. | This results from [[Euler's identity]] {{tmath|1=e^{i\pi}=-1}} and the functional identity. | ||
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<math display="block">\overline{e^z}=e^{\overline z}.</math> | <math display="block">\overline{e^z}=e^{\overline z}.</math> | ||
Its modulus is | Its modulus is | ||
<math display="block">|e^z|= e^{ | <math display="block">|e^z|= e^{\Re (z)},</math> | ||
where {{tmath|\Re(z)}} denotes the real part of {{tmath|z}}. | where {{tmath|\Re(z)}} denotes the real part of {{tmath|z}}. | ||
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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"/> | 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=== | === 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"> | <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"> | ||
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<gallery class="center" mode="packed" style="text-align:left" heights="200px" caption="Graphs of the complex exponential function"> | <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: | 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 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 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 | 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> | </gallery> | ||
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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 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 | 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 {{math|2''π''}} periodicity in the imaginary <math>y</math> value. | ||
== | ==Transcendency== | ||
The | 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 [[field (mathematics)|field]] of the [[rational fraction]]s <math>\C(z);</math> in fact, this is true for any exponential function with a positive real base not equal to 1. | ||
This follows from the stronger statement that 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>. | |||
A much more difficult result is that the base ''e'' of the natural exponential function is a [[transcendental number]], see the [[Lindemann–Weierstrass theorem]]. | |||
=={{anchor|exp|expm1}}Computation== | =={{anchor|exp|expm1}}Computation== | ||
| Line 300: | Line 301: | ||
<math display="block"> e^x = 1 + \cfrac{x}{1 - \cfrac{x}{x + 2 - \cfrac{2x}{x + 3 - \cfrac{3x}{x + 4 - \ddots}}}}</math> | <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"/> | The following [[generalized continued fraction]] for {{math|''e''{{isup|''z''}}}}, also due to Euler | ||
,<ref>A. N. Khovanski, The applications of continued fractions and their generalization to problems 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> | <math display="block"> e^z = 1 + \cfrac{2z}{2 - z + \cfrac{z^2}{6 + \cfrac{z^2}{10 + \cfrac{z^2}{14 + \ddots}}}}</math> | ||
| Line 311: | Line 314: | ||
<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"> 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== | == Generalizations == | ||
{{ | ===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}\to G</math> 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 [[general linear 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. | |||
== See also == | |||
{{portal|Mathematics}} | |||
{{div col}} | {{div col}} | ||
* [[Carlitz exponential]], a characteristic {{math|''p''}} analogue | * [[Carlitz exponential]], a characteristic {{math|''p''}} analogue | ||
| Line 319: | Line 336: | ||
* [[Gaussian function]] | * [[Gaussian function]] | ||
* [[Half-exponential function]], a compositional square root of an exponential function | * [[Half-exponential function]], a compositional square root of an exponential function | ||
* {{annotated link|Lambert W function#Solving equations}} | * {{annotated link|Lambert W function#Solving equations}} – used for solving exponential equations | ||
* [[List of exponential topics]] | * [[List of exponential topics]] | ||
* [[List of integrals of exponential functions]] | * [[List of integrals of exponential functions]] | ||
| Line 332: | Line 349: | ||
==References== | ==References== | ||
<references> | |||
<!-- <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_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="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> | ||
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</ref> | </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="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="Apostol_1974">{{Cite book |title=Mathematical Analysis |url=https://archive.org/details/mathematicalanal00apos_530 |url-access=limited |author-last=Apostol |author-first=Tom M. |author-link=Tom M. Apostol |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_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="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> | ||
| Line 347: | Line 364: | ||
<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="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> | <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> | ||
</references> | |||
==External links== | ==External links== | ||
Latest revision as of 20:34, 24 April 2026
Template:Infobox mathematical function
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted Template:Tmath or Template:Tmath; the latter is preferred when the argument Template:Tmath is a complicated expression.[1][2] It is called exponential because its argument can be seen as an exponent to which a constant number e ≈ 2.718, the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.
The exponential function converts sums to products: Template:Tmath. Its inverse function, the natural logarithm, Template:Tmath or Template:Tmath, converts products to sums: Template:Tmath.
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 Template:Tmath, which is exponentiation with a fixed base Template:Tmath. More generally, and especially in applications, functions of the general form Template:Tmath are also called exponential functions. They grow or decay exponentially in that the rate that Template:Tmath changes when Template:Tmath is increased is proportional to the current value of Template:Tmath.
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 Template:Tmath 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
[edit | edit source]The graph of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=e^x} is upward-sloping, and increases faster than every power of Template:Tmath.[3] 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{d}{dx}e^x = e^x} means that the slope of the tangent to the graph at each point is equal to its height (its y-coordinate) at that point.
Definitions and fundamental properties
[edit | edit source]There are several equivalent definitions of the exponential function, although of very different nature.
Differential equation
[edit | edit source]The exponential function is the unique differentiable function that equals its derivative, and takes the value 1 for the value 0 of its variable.
This definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.
Inverse of natural logarithm
[edit | edit source]The exponential function is the inverse function of the natural logarithm. That is,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \ln (\exp x)&=x\\ \exp(\ln y)&=y \end{align}}
for every real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and every positive real number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y.}
Power series
[edit | edit source]The exponential function is the sum of the power series[4][5] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}\exp(x) &= 1+x+\frac{x^2}{2!}+ \frac{x^3}{3!}+\cdots\\ &=\sum_{n=0}^\infty \frac{x^n}{n!},\end{align}}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n!} is the factorial of n (the product of the n first positive integers). This series is absolutely convergent for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , by the ratio test. This shows that the exponential function is defined for every Template:Tmath, and is everywhere the sum of its Maclaurin series.
Functional equation
[edit | edit source]The exponential satisfies the functional equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp(x+y)= \exp(x)\cdot \exp(y)} and maps the additive identity 0 to the multiplicative identity 1. The same equation is satisfied by other continuous functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=b^x} that exponentiate their argument with an arbitrary base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} .[6] Among these functions, the exponential function is characterized by the property that its derivative at 0 is 1.[7]
Limit of integer powers
[edit | edit source]The exponential function is the limit, as the integer n goes to infinity,[8][5] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp(x)=\lim_{n \to +\infty} \left(1+\frac xn\right)^n.}
Properties
[edit | edit source]Reciprocal: The functional equation implies Template:Tmath. Therefore Template:Tmath for every Template:Tmath and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac 1{e^x}=e^{-x}.}
Positiveness: Template:Tmath for every real number Template:Tmath. This results from the intermediate value theorem, since Template:Tmath and, if one would have Template:Tmath for some Template:Tmath, there would be an Template:Tmath such that Template:Tmath between Template:Tmath and Template:Tmath. 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(\ln b).} If n is an integer, the functional equation of the logarithm implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Template:Tmath for every positive real number b and every real number x: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^x=\exp(x\ln b).} In particular, if b is the Euler's number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e=\exp(1),} one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln e=1} (inverse function) and thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^x=\exp(x).} This shows the equivalence of the two notations for the exponential function.
General exponential functions
[edit | edit source]A function is commonly called an exponential function—with an indefinite article—if it has the form Template:Tmath, 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 Template:Tmath. 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 Template:Tmath, and so, it is nonsensical to impose Template:Tmath.
These most general exponential functions are the differentiable functions that satisfy the following equivalent characterizations.
- Template:Tmath for every Template:Tmath and some constants Template:Tmath and Template:Tmath.
- Template:Tmath for every Template:Tmath and some constants Template:Tmath and Template:Tmath.
- The value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)/f(x)} is independent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .
- For every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d,} the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x+d)/f(x)} is independent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x;} that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(x+d)}{f(x)}= \frac{f(y+d)}{f(y)}} for every x, y.[9]
The base of an exponential function is the base of the exponentiation that appears in it when written as Template:Tmath, namely Template:Tmath.[10] The base is Template:Tmath in the second characterization, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \exp \frac{f'(x)}{f(x)}} in the third one, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \left(\frac{f(x+d)}{f(x)}\right)^{1/d}} in the last one.
In applications
[edit | edit source]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 Template:Tmath or, equivalently, is a solution of the differential equation Template:Tmath, the constant Template:Tmath is called, depending on the context, the decay constant, disintegration constant,[11] rate constant,[12] or transformation constant.[13]
Equivalence proof
[edit | edit source]For proving the equivalence of the above properties, one can proceed as follows.
The two first characterizations are equivalent, since, if Template:Tmath and Template:Tmath, one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{kx}= (e^k)^x= b^x.} The basic properties of the exponential function (derivative and functional equation) implies immediately the third and the last condition.
Suppose that the third condition is verified, and let Template:Tmath be the constant value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)/f(x).} Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \frac {\partial e^{kx}}{\partial x}=ke^{kx},} the quotient rule for derivation implies that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac \partial{\partial x}\,\frac{f(x)}{e^{kx}}=0,} and thus that there is a constant Template:Tmath such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=ae^{kx}.}
If the last condition is verified, let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \varphi(d)=f(x+d)/f(x),} which is independent of Template:Tmath. Using Template:Tmath, one gets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{f(x+d)-f(x)}{d} = f(x)\,\frac{\varphi(d)-\varphi(0)}{d}. } Taking the limit when Template:Tmath tends to zero, one gets that the third condition is verified with Template:Tmath. It follows therefore that Template:Tmath for some Template:Tmath and Template:Tmath As a byproduct, one gets that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{f(x+d)}{f(x)}\right)^{1/d}=e^k} is independent of both Template:Tmath and Template:Tmath.
Compound interest
[edit | edit source]The earliest occurrence of the exponential function was in Jacob Bernoulli's study of compound interests in 1683.[14] This is this study that led Bernoulli to consider the number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^{n}} now known as Euler's number and denoted Template:Tmath.
The exponential function is involved as follows in the computation of continuously compounded interests.
If a principal amount of 1 earns interest at an annual rate of x compounded monthly, then the interest earned each month is x/12 times the current value, so each month the total value is multiplied by (1 + x/12), and the value at the end of the year is (1 + x/12)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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp x = \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^{n}} first given by Leonhard Euler.[8]
Differential equations
[edit | edit source]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 Template:Tmath. Every other exponential function, of the form Template:Tmath, is a solution of the differential equation Template:Tmath, and every solution of this differential equation has this form.
The solutions of an equation of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'+ky=f(x)} involve exponential functions in a more sophisticated way, since they have the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=ce^{-kx}+e^{-kx}\int f(x)e^{kx}dx,} where Template:Tmath 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
[edit | edit source]
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 Template:Tmath or Template:Tmath. 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 Template:Tmath for the argument Template:Tmath: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{de^z}{dz}=e^z\quad\text{and}\quad e^0=1.}
The complex exponential function is the sum of the series Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^z = \sum_{k = 0}^\infty\frac{z^k}{k!}.} This series is absolutely convergent for every complex number Template:Tmath. So, the complex exponential is an entire function.
The complex exponential function is the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^z = \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n}
As with the real exponential function (see § Functional equation above), the complex exponential satisfies the functional equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp(z+w)= \exp(z)\cdot \exp(w).} Among complex functions, it is the unique solution which is holomorphic at the point Template:Tmath and takes the derivative Template:Tmath there.[15]
The complex logarithm is a right-inverse function of the complex exponential: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{\log z} =z. } However, since the complex logarithm is a multivalued function, one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log e^z= \{z+2ik\pi\mid 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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac 1{e^z}=e^{-z} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^z\neq 0\quad \text{for every } z\in \C .} It is periodic function of period Template:Tmath; that is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{z+2ik\pi} =e^z \quad \text{for every } k\in \Z.} This results from Euler's identity Template:Tmath and the functional identity.
The complex conjugate of the complex exponential is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{e^z}=e^{\overline z}.} Its modulus is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |e^z|= e^{\Re (z)},} where Template:Tmath denotes the real part of Template:Tmath.
Relationship with trigonometry
[edit | edit source]Complex exponential and trigonometric functions are strongly related by Euler's formula: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{it} =\cos(t)+i\sin(t). }
This formula provides the decomposition of complex exponentials into real and imaginary parts: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{x+iy} = e^{x}e^{iy} = e^x\,\cos y + i e^x\,\sin y.}
The trigonometric functions can be expressed in terms of complex exponentials: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \cos x &= \frac{e^{ix}+e^{-ix}}2\\ \sin x &= \frac{e^{ix}-e^{-ix}}{2i}\\ \tan x &= i\,\frac{1-e^{2ix}}{1+e^{2ix}} \end{align}}
In these formulas, Template:Tmath 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.[16]
Plots
[edit | edit source]- 3D plots of real part, imaginary part, and modulus of the exponential function
-
z = Re(eTemplate:Isup)
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z = Im(eTemplate:Isup)
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z = |eTemplate:Isup|
Considering the complex exponential function as a function involving four real variables: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} domain, the following are depictions of the graph as variously projected into two or three dimensions.
- Graphs of the complex exponential function
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Checker board key: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x> 0:\; \text{green}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x< 0:\; \text{red}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y> 0:\; \text{yellow}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y< 0:\; \text{blue}}
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Projection onto the range complex plane (V/W). Compare to the next, perspective picture.
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Projection into the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image)
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Projection into the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} dimensions, producing a spiral shape (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} range extended to ±2π, 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} axis is mapped to the positive real Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} axis
- the imaginary Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} axis. It shows the graph is a surface of revolution about the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} axis. It shows that the graph's surface for positive and negative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} values doesn't really meet along the negative real Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} axis, but instead forms a spiral surface about the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} axis. Because its Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} values have been extended to ±2π, this image also better depicts the 2π periodicity in the imaginary Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} value.
Transcendency
[edit | edit source]The function eTemplate:Isup is a transcendental function, which means that it is not a root of a polynomial over the field of the rational fractions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \C(z);} in fact, this is true for any exponential function with a positive real base not equal to 1.
This follows from the stronger statement that if a1, ..., an are distinct complex numbers, then ea1z, ..., eanz are linearly independent over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \C(z)} .
A much more difficult result is that the base e of the natural exponential function is a transcendental number, see the Lindemann–Weierstrass theorem.
Computation
[edit | edit source]The Taylor series definition above is generally efficient for computing (an approximation of) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^x} . However, when computing near the argument Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0} , the result will be close to 1, and computing the value of the difference Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 ex − 1 directly, bypassing computation of eTemplate:Isup. For example,
one may use the Taylor series:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^x-1=x+\frac {x^2}2 + \frac{x^3}6+\cdots +\frac{x^n}{n!}+\cdots.}
This was first implemented in 1979 in the Hewlett-Packard HP-41C calculator, and provided by several calculators,[17][18] operating systems (for example Berkeley UNIX 4.3BSD[19]), computer algebra systems, and programming languages (for example C99).[20]
In addition to base e, the IEEE 754-2008 standard defines similar exponential functions near 0 for base 2 and 10: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^x - 1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^x - 1} .
A similar approach has been used for the logarithm; see log1p.
An identity in terms of the hyperbolic tangent, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{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
[edit | edit source]The exponential function can also be computed with continued fractions.
A continued fraction for eTemplate:Isup can be obtained via an identity of Euler: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^x = 1 + \cfrac{x}{1 - \cfrac{x}{x + 2 - \cfrac{2x}{x + 3 - \cfrac{3x}{x + 4 - \ddots}}}}}
The following generalized continued fraction for eTemplate:Isup, also due to Euler ,[21] converges more quickly:[22] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^z = 1 + \cfrac{2z}{2 - z + \cfrac{z^2}{6 + \cfrac{z^2}{10 + \cfrac{z^2}{14 + \ddots}}}}}
or, by applying the substitution z = x/y: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^\frac{x}{y} = 1 + \cfrac{2x}{2y - x + \cfrac{x^2} {6y + \cfrac{x^2} {10y + \cfrac{x^2} {14y + \ddots}}}}} with a special case for z = 2: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 }}}}}
This formula also converges, though more slowly, for z > 2. For example: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 }}}}}
Generalizations
[edit | edit source]Matrices and Banach algebras
[edit | edit source]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, eTemplate:Isup = 1, and eTemplate:Isup is invertible with inverse eTemplate:Isup for any x in B. If xy = yx, then eTemplate:Isup = eTemplate:IsupeTemplate:Isup, but this identity can fail for noncommuting x and y.
Some alternative definitions lead to the same function. For instance, eTemplate:Isup can be defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} \left(1 + \frac{x}{n} \right)^n .}
Or eTemplate:Isup can be defined as fx(1), where fx : R → B is the solution to the differential equation dfx/dt(t) = x fx(t), with initial condition fx(0) = 1; it follows that fx(t) = eTemplate:Isup for every t in R.
Lie algebras
[edit | edit source]Given a Lie group G and its associated Lie algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}} , the exponential map is a map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{g}\to 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 × n matrices has as Lie algebra M(n,R), the space of all n × n matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.
The identity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.
See also
[edit | edit source]- Carlitz exponential, a characteristic p analogue
- Double exponential function
- Exponential field
- Gaussian function
- Half-exponential function, a compositional square root of an exponential function
- Lambert W function#Solving equations – Multivalued function in mathematics – 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
- Padé table for exponential function – Padé approximation of exponential function by a fraction of polynomial functions
- Phase factor
Notes
[edit | edit source]References
[edit | edit source]- ↑ "Reviews of Modern Physics Style Guide" (PDF). XVI.B.1(d): American Physical Society. p. 18. Retrieved 2025-12-30.
Which form to use, Template:Tmath or Template:Tmath, is determined by the number of characters and the complexity of the argument. The Template:Tmath form is appropriate when the argument is short and simple, i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i \mathbf{k}\cdot\mathbf{r}}} , whereas Template:Tmath should be used if the argument is more complicated.
CS1 maint: location (link) - ↑ T. W. Chaundy; P. R. Barrett; Charles Batey (1954). The Printing of Mathematics. Oxford University Press. p. 31.
- ↑ "Exponential Function Reference". www.mathsisfun.com. Retrieved 2020-08-28.
- ↑ Rudin, Walter (1987). Real and complex analysis (3rd ed.). New York: McGraw-Hill. p. 1. ISBN 978-0-07-054234-1.
- ↑ 5.0 5.1 Weisstein, Eric W. "Exponential Function". mathworld.wolfram.com. Retrieved 2020-08-28.
- ↑ Jung, Soon-Mo (2011). "Chapter 9: Exponential Functional Equations". Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer Optimization and Its Applications. 48. Springer New York. pp. 207–225. doi:10.1007/978-1-4419-9637-4_9. ISBN 9781441996374.
- ↑ Aczél, J.; Dhombres, J. (1989). Functional Equations in Several Variables. Encyclopedia of Mathematics and its Applications. 31. Cambridge University Press, Cambridge. p. 10. doi:10.1017/CBO9781139086578. ISBN 0-521-35276-2. MR 1004465.
- ↑ 8.0 8.1 Maor, Eli. 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. LCCN 75173716.
- ↑ 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
- ↑ Hille, Einar (1959). "The exponential function". Analytic Function Theory. 1. Waltham, MA: Blaisdell. § 6.1, Template:Pgs.
- ↑ Apostol, Tom M. (1974). Mathematical Analysis (2nd ed.). Reading, Mass.: Addison Wesley. pp. 19. ISBN 978-0-201-00288-1.
- ↑ HP 48G Series – Advanced User's Reference Manual (AUR) (4 ed.). Hewlett-Packard. December 1994 [1993]. HP 00048-90136, 0-88698-01574-2. Retrieved 2015-09-06.
- ↑ HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR) (2 ed.). Hewlett-Packard. 2009-07-14 [2005]. HP F2228-90010. Retrieved 2015-10-10. [1]
- ↑ Beebe, Nelson H. F. (2017-08-22). "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. pp. 273–282. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. S2CID 30244721.
Berkeley UNIX 4.3BSD introduced the expm1() function in 1987.
- ↑ Beebe, Nelson H. F. (2002-07-09). "Computation of expm1 = exp(x)−1" (PDF). 1.00. Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing, University of Utah. Retrieved 2015-11-02.
- ↑ A. N. Khovanski, The applications of continued fractions and their generalization to problems in approximation theory,1963, Noordhoff, Groningen, The Netherlands
- ↑ Lorentzen, L.; Waadeland, H. (2008). "A.2.2 The exponential function.". Continued Fractions. Atlantis Studies in Mathematics. 1. p. 268. doi:10.2991/978-94-91216-37-4. ISBN 978-94-91216-37-4.