Exponential function: Difference between revisions

Revision as of 13:07, 11 February 2025

Latest revision as of 07:29, 22 August 2025

(48 intermediate revisions by 26 users not shown)

Line 23:

}}

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 amountrate 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.

Line 66:

&=\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 derivationdifferentiation, 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

Line 77:

===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>

where <math>n</math> takes only integer values (otherwise, the exponentiation would require the exponential function to be defined). 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]].

Line 114:

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 varaiblevariable 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 popertiesproperties, 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

s.<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 thsthe last condititoncondition.

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

Line 160:

where {{tmath|c}} is an arbitrary constant and the integral denotes any [[antiderivative]] of its argument.

More geneallygenerally, 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 <span class="anchor" id="On the complex plane"></span><span class="anchor" id="Complex plane"></span> ==

{{see alsoanchor|Euler'sOn formula#Definitions ofthe complex exponentiationplane|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 − 2-2i''i''}} to {{math|2 +2i 2''i''}}|thumb|The exponential function {{math|''e^''{{isup|''z''}}}} plotted in the complex plane from -{{math|−2 − 2-2i''i''}} to {{math|2 +2i 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'''.

As in the [[real number|real]] case, the exponential function can be defined on the [[complex plane]] in several equivalent forms.

TheMost mostof commonthe definitiondefinitions of the complex exponential function parallelscan the powerbe seriesused definitionverbatim for realdefiniting argumentsthe complex exponential function, whereand the realproof of their variableequivalence is replacedthe bysame aas complexin the real one:case.

<math display="block">\exp z := \sum_{k = 0}^\infty\frac{z^k}{k!}</math>

Alternatively, theThe complex exponential function maycan be defined by modellingin theseveral limitequivalent definitionways forthat realare arguments,the butsame withas in the real variable replaced by a complex one:case.

<math display="block">\exp z := \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n</math>

The ''complex exponential'' is the unique complex function that equals its [[complex derivative]] and takes the value {{tmath|1}} for the argument {{tmath|0}}:

For the power series definition, term-wise multiplication of two copies of this power series in the [[Cauchy product|Cauchy]] sense, permitted by [[Cauchy product|Mertens' theorem]], shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments:

<math display="block">\exp(w+frac{de^z)}{dz}=e^z\exp w\exp z quad\text { for all and}\quad w,z\in\mathbb{C}e^0=1.</math>

The definition of the ''complex exponential function'' in turn leads tois the appropriatesum definitions extendingof the [[trigonometricseries functions(mathematics)|series]] to complex arguments.

<math display="block">\exp 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]]

In particular, when {{math|1=''z'' = ''it''}} ({{mvar|t}} real), the series definition yields the expansion

<math display="block">\exp(it)e^z = \left( 1-\fraclim_{t^2}{2!}+n\frac{t^4}{4!}-to\frac{t^6infty}{6!}+\cdots \right) + i\left(t - \frac{t^3}{3!} 1+ \frac{t^5z}{5!n} - \frac{t^7}{7!}+\cdots\right).^n</math>

As with the real exponential function (see {{slink||Functional equation}} above), the complex exponential satisfies the functional equation

In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series. The real and imaginary parts of the above expression in fact correspond to the series expansions of {{math|cos ''t''}} and {{math|sin ''t''}}, respectively.

<math display="block">\exp(z^+w )= e^{w \log 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:

This correspondence provides motivation for {{em|defining}} cosine and sine for all complex arguments in terms of <math>\exp(\pm iz)</math> and the equivalent power series:<ref name="Rudin_1976"/>

<math display="block">\begine^{align\log z} =z. </math>

However, since the complex logarithm is a [[multivalued function]], one has

& \cos z:= \frac{\exp(iz)+\exp(-iz)}{2} = \sum_{k=0}^\infty (-1)^k \frac{z^{2k}}{(2k)!}, \\[5pt]

<math display="block">\explog e^z := \lim_{nz+2ik\topi\infty}mid k\left(1+in \Z\frac{z}{n}\right)^n,</math>

\text{and } \quad & \sin z := \frac{\exp(iz)-\exp(-iz)}{2i} =\sum_{k=0}^\infty (-1)^k\frac{z^{2k+1}}{(2k+1)!}

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.

\end{align}</math>

The complex exponential has the following properties:

for all <math display=inline"block">\frac 1{e^z\in\mathbb}=e^{C-z}. </math>

and

<math display="block">e^z\exp(iz)=\cosneq z+i0\sin zquad \text { for allevery } z\in \mathbb{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

The functions {{math|exp}}, {{math|cos}}, and {{math|sin}} so defined have infinite [[Radius of convergence|radii of convergence]] by the [[ratio test]] and are therefore [[entire function]]s (that is, [[Holomorphic function|holomorphic]] on <math>\mathbb{C}</math>). The range of the exponential function is <math>\mathbb{C}\setminus \{0\}</math>, while the ranges of the complex sine and cosine functions are both <math>\mathbb{C}</math> in its entirety, in accord with [[Picard theorem|Picard's theorem]], which asserts that the range of a nonconstant entire function is either all of <math>\mathbb{C}</math>, or <math>\mathbb{C}</math> excluding one [[lacunary value]].

<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===

These definitions for theComplex exponential and [[trigonometric functionsfunction]]s are leadstrongly triviallyrelated toby [[Euler's formula]]:

<math display="block">\exp(iz)=\cos z+i\sin z \text { for all } z\in\mathbb{C}.</math>

<math display="block">\int_0e^{t_0it}| =\gamma'cos(t)| \, dt = \int_0^{t_0} |+i\expsin(itt)| \, dt =. t_0,</math>

This formula provides the decomposition of complex exponentials into [[real and imaginary parts]]:

We could alternatively define the complex exponential function based on this relationship. If {{math|1=''z'' = ''x'' + ''iy''}}, where {{mvar|x}} and {{mvar|y}} are both real, then we could define its exponential as

<math display="block">\exp ze^{x+iy} = \exp(e^{x+}e^{iy)} := (\exp e^x)(\,\cos y + i e^x\,\sin y).</math>

where {{math|exp}}, {{math|cos}}, and {{math|sin}} on the right-hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means.<ref name="Apostol_1974"/>

The trigonometric functions can be expressed in terms of complex exponentials:

For <math>t\in\R</math>, the relationship <math>\overline{\exp(it)}=\exp(-it)</math> holds, so that <math>\left|\exp(it)\right| = 1</math> for real <math>t</math> and <math>t \mapsto \exp(it)</math> maps the real line (mod {{math|2''π''}}) to the [[unit circle]] in the complex plane. Moreover, going from <math>t = 0</math> to <math>t = t_0</math>, the curve defined by <math>\gamma(t)=\exp(it)</math> traces a segment of the unit circle of length

<math display="block">\int_0^{t_0}|\gamma'(t)| \, dt = \int_0^{t_0} |i\exp(it)| \, dt = t_0,</math>

starting from {{math|1=''z'' = 1}} in the complex plane and going counterclockwise. Based on these observations and the fact that the measure of an angle in radians is the arc length on the unit circle subtended by the angle, it is easy to see that, restricted to real arguments, the sine and cosine functions as defined above coincide with the sine and cosine functions as introduced in elementary mathematics via geometric notions.

The complex exponential function is periodic with period {{math|2''πi''}} and <math>\exp(z+2\pi i k)=\exp z</math> holds for all <math>z \in \mathbb{C}, k \in \mathbb{Z}</math>.

When its ___domain is extended from the real line to the complex plane, the exponential function retains the following properties:

<math display="block">\begin{align}

\cos x &= \frac{e^{z ix}+ w} = e^z e^w {-ix}}2\\[5pt]

\sin x &= \frac{e^{ix}-e^{-ix}}{2i}\\

& e^0 = 1 \\[5pt]

\tan x &= i\,\frac{1-e^{2ix}}{1+e^{2ix}}

& e^z \ne 0 \\[5pt]

\end{align}</math>

& \frac{d}{dz} e^z = e^z \\[5pt]

& \left(e^z\right)^n = e^{nz}, n \in \mathbb{Z}

\end{align}</math>

for all <math display=inline> w,z\in\mathbb C.</math>

Extending the natural logarithm to complex arguments yields the [[complex logarithm]] {{math|log ''z''}}, which is a [[multivalued function]].

We can then define a more general exponentiation:

<math display="block">z^w = e^{w \log z}</math>

for all complex numbers {{math|''z''}} and {{math|''w''}}. This is also a multivalued function, even when {{math|''z''}} is real. This distinction is problematic, as the multivalued functions {{math|log ''z''}} and {{math|''z''^''w''}} are easily confused with their single-valued equivalents when substituting a real number for {{math|''z''}}. The rule about multiplying exponents for the case of positive real numbers must be modified in a multivalued context:

{{block indent|em=1.5|text={{math|(''e{{sup|z}}''){{su|p=''w''}} ≠ ''e{{sup|zw}}''}}, but rather {{math|1=(''e{{sup|z}}''){{su|p=''w''}} = ''e''{{sup|(''z'' + 2''niπ'')''w''}}}} multivalued over integers {{math|''n''}}}}

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"/>

See [[Exponentiation#Failure of power and logarithm identities|failure of power and logarithm identities]] for more about problems with combining powers.

===Plots===

The exponential function maps any [[line (mathematics)|line]] in the complex plane to a [[logarithmic spiral]] in the complex plane with the center at the [[Origin (mathematics)|origin]]. Two special cases exist: when the original line is parallel to the real axis, the resulting spiral never closes in on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius.

<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>

Line 265 ⟶ 260:

==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''_''x''(1)}}, where {{math|''f''_''x'' : '''R''' → ''B''}} is the solution to the differential equation {{math|1={{sfrac|''df''_''x''''|''dt''}}(''t'') = ''x{{space|hair}}f''_''x''(''t'')}}, with initial condition {{math|1=''f''_''x''(0) = 1}}; it follows that {{math|1=''f''_''x''(''t'') = ''e''^{{{isup|''tx''}}}}} for every {{mvar|t}} in {{math|'''R'''}}.

==Lie algebras==

Line 278 ⟶ 273:

==Transcendency==

The function {{math|''e''^{{isup|''z''}}}} is not ina the rational[[transcendental function]], ringwhich <math>\C(z)</math>:means that it is not thea [[polynomial quotientroot|root]] of twoa polynomialspolynomial withover complexthe coefficients[[ring (mathematics)|ring]] of the [[rational fraction]]s <math>\C(z).</math>

If {{math|''a''₁, ..., ''a''_''n''<nowiki/>}} are distinct complex numbers, then {{math|''e''^{''a''₁''z''}, ..., ''e''^{''a''_''n''''z''}<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^x'' − 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>

Line 302 ⟶ 297:

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>

Line 338 ⟶ 333:

==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>