Exponential function: Difference between revisions

| image = Image:exp.svg

| derivative = <math>\exp'\! z = \exp z</math>

In [[mathematics]], the '''exponential function''' is the unique [[real function]] which maps [[0|zero]] to [[1|one]] and has a [[derivative (mathematics)|derivative]] everywhere equal to its value. The exponential of a variable {{tmath|x}} is denoted {{tmath|\exp x}} or {{tmath|e^x}}, with the two notations used interchangeably. It is called ''exponential'' because its argument can be seen as an [[exponent (mathematics)|exponent]] to which a constant [[e (mathematical constant)|number {{math|''e'' ≈ 2.718}}]], the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.

The exponential function converts sums to products: it maps the [[additive identity]] {{math|0}} to the [[multiplicative identity]] {{math|1}}, and the exponential of a sum is equal to the product of separate exponentials, {{tmath|1=\exp(x + y) = \exp x \cdot \exp y }}. Its [[inverse function]], the [[natural logarithm]], {{tmath|\ln}} or {{tmath|\log}}, converts products to sums: {{tmath|1= \ln(x\cdot y) = \ln x + \ln y}}.

The exponential function 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.

One of the simplest definitions is: The ''exponential function'' is the ''unique'' [[differentiable function]] that equals its [[derivative]], and takes the value {{math|1}} for the value {{math|0}} of its variable.

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

 

 

''Existence'' is proved in each of the two following sections.

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

 

''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>  

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

''The exponential satisfies the functional equation:''

<math display=block>\exp(x+y)= \exp(x)\cdot \exp(y).</math>

This results from the uniqueness and the fact that the function

<math> f(x)=\exp(x+y)/\exp(y)</math> satisfies the above definition.  

 

It can be proved that a function that satisfies this functional equation has the form {{tmath|x \mapsto \exp(cx)}} if it is either [[continuous function|continuous]] or [[monotonic function|monotonic]]. It is thus [[differentiable function|differentiable]], and equals the exponential function if its derivative at {{math|0}} is {{math|1}}.

''The exponential function is the [[limit (mathematics)|limit]], as the integer {{mvar|n}} goes to infinity,<ref name="Maor"/><ref name=":0" />

''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  

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.

The last characterization is important in [[empirical science]]s, as allowing a direct [[experimental]] test whether a function is an exponential function.  

<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>  

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

==Complex exponential==

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'''.

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

This series is [[absolutely convergent]] for every complex number {{tmath|z}}. So, the complex differential is an [[entire function]].

The functional equation

<math display="block">e^{w+z}=e^we^z</math>

holds for every complex numbers {{tmath|w}} and {{tmath|z}}. The complex exponential is the unique [[continuous function]] that satisfies this functional equation and has the value {{tmath|1}} for {{tmath|1=z=0}}.

The [[complex logarithm]] is a [[left inverse function|right-inverse function ]] of the complex exponential:

It is [[periodic function|periodic function]] of period {{tmath|2i\pi}}; that is  

<math display="block">|e^z|= e^{|\Re (z)|},</math>

===Plots===

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 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).

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

==Matrices and Banach algebras==

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

 

Some alternative definitions lead to the same function.  For instance, {{math|''e''{{isup|''x''}}}} can be defined as

 

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'''}}.

 

Given a [[Lie group]] {{math|''G''}} and its associated [[Lie algebra]] <math>\mathfrak{g}</math>, the [[exponential map (Lie theory)|exponential map]] is a map <math>\mathfrak{g}</math> {{math|↦ ''G''}} satisfying similar properties. In fact, since {{math|'''R'''}} is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie group {{math|GL(''n'','''R''')}} of invertible {{math|''n'' × ''n''}} matrices has as Lie algebra {{math|M(''n'','''R''')}}, the space of all {{math|''n'' × ''n''}} matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.

 

The identity <math>\exp(x+y)=\exp(x)\exp(y)</math> can fail for Lie algebra elements {{math|''x''}} and {{math|''y''}} that do not commute; the [[Baker–Campbell–Hausdorff formula]] supplies the necessary correction terms.

The function {{math|''e''{{isup|''z''}}}} is  a [[transcendental function]], which means that it is not a [[polynomial root|root]] of a polynomial over the [[ring (mathematics)|ring]] of the [[rational fraction]]s <math>\C(z).</math>

If {{math|''a''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>.

The following [[generalized continued fraction]] for {{math|''e''{{isup|''z''}}}} converges more quickly:<ref name="Lorentzen_2008"/>

==See also==

{{Portal|Mathematics}}

* {{annotated link|Lambert W function#Solving equations}} - Used for solving exponential equations

<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>