Exponential function: Difference between revisions

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[[Category:Complex analysis]]

The '''exponential function''' is one of the most important [[function (mathematics)|function]]s in [[mathematics]]. It is written as exp(''x'') or <i>''e''^''x''</i>, (where <i>''e</i>'' is the [[e (mathematical constant)|base of the natural logarithm]]).

[[image:exp.png|right|The exponential function is nearly flat (climbing slowly) for negative x's, and climbs quickly for positive x's.]]<br>

The graph of <font color=#803300><i>e^x</i></font> is always increasing. It does '''not''' ever touch the <i>x</i> axis, although it comes arbitrarily close.

As a function of the ''[[real number|real]]'' variable ''x'', the [[graph of a function|graph]] of <font color=#803300>''e''^''x''</font> is always positive (above the ''x'' axis) and increasing (viewed left-to-right). It never touches the ''x'' axis, although it gets arbitrarily close to it (thus, the ''x'' axis is a horizontal [[asymptote]] to the graph). Its [[inverse function]], the [[natural logarithm]], ln(''x''), is defined for all positive ''x''.

It can be defined in two equivalent ways: an [[infinite series]] or a [[limit]] of a sequence:

Sometimes, especially in the [[science]]s, the term '''exponential function''' is reserved for functions of the form ''ka''^''x'',

:<math>e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots</math>

where ''a'', called the ''base'', is any positive real number. This article will focus initially on the exponential function with base ''e''.

:<math>e^x = \lim_{n \to \infty} \left( 1 + {x \over n} \right)^n</math>

HereIn general, <math>n!</math> stands for the [[factorialvariable]] of <i>n</i> and <i>''x</i>'' can be any [[real number|real]] or [[complex number|complex]] number, or even anyan entirely elementdifferent kind of amathematical [[Banachobject; algebra]] orsee the field of [[p-adic#Formal numbersdefinition|<i>p</i>-adicformal definition numbersbelow]].

==Properties==

To see the equivalence of these definitions, see [[Definitions of the exponential function]].

If <i>x</i> is real, then <i>e^x</i> is positive and strictly increasing. Therefore its [[inverse function]], the [[natural logarithm]] ln(<i>x</i>), is defined for all positive <i>x</i>. Using the natural logarithm, one can define more general exponential functions. asThe follows:function

: <math>a^x=e^{x \ln a}</math>

defined for all ''a'' > 0, and all real numbers ''x'', is called the '''exponential function with base''' '''''a'''''.

for all <i>a</i> > 0 and <math>x \in \mathbb{R}</math>.

Note that the equation above holds for ''a'' = ''e'', since

This works for <i>a</i>=<i>e</i>:

: <math>\mbox{Right hand side}=e^{x \ln e}=e^{x\left(1\right)}=e^x=\mbox{Left hand side}.</math>

The exponential function also gives rise to the [[trigonometric function]]s (as can be seen from [[Eulers formula in complex analysis|Euler's formula]]) and to the [[hyperbolic function]]s. Thus we see that all elementary functions except for the [[polynomial]]s spring from the exponential function in one way or another.

Exponential functions "translate between addition and multiplication" as is expressed in the following ''exponential laws'':

: <math>a^0 = 1</math>

: <math>a^1 = a</math>

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: <math>a^x b^x = (a b)^x</math>

These are valid for all positive real numbers <i>''a</i>'' and <i>''b</i>'' and all real numbers <i>''x</i>'' and ''y''. Expressions involving fractions[[fraction]]s and [[Radical (mathematics)|roots]] can often be simplified using exponential notation because:

: <math>{1 \over a} = a^{-1}</math>

and, for any ''a'' > 0, real number ''b'', and integer ''n''&nbsp;>&nbsp;1:

: <math>\sqrt[cn]{a}^b} = \left(\sqrt[n]{a}\right)^{b \over= ca^{b/n}</math>

==Exponential functionDerivatives and differential equations==

The major importance of the exponential functions in mathematics and the sciences stems mainly from the fact that they are constant multiplesproperties of their own [[Derivative|derivativesderivative]]:s. In particular,

: <math>{dyd \over dx} e^x = ye^x</math>

That is, ''e''^''x'' is its own [[derivative]], a property unique among real-valued functions of a real variable. Other ways of saying the same thing include:

*The slope of the graph at any point is the height of the function at that point.

*The rate of increase of the function at ''x'' is equal to the value of the function at ''x''.

*The exponential function thus solves the basic [[differential equation]] ''y''&prime; = ''y''.

andIn itfact, is for this reason commonly encountered inmany differential equations. give Inrise particular the solution of linear ordinary [[differential equation]]s can frequently be written in terms ofto exponential functions., including These equations includethe [[Schrödinger equation]] and the [[Laplace's equation]] as well as the equations for [[simple harmonic motion]].

For exponential functions with other bases:

: <math>{d \over dx} a^x = (\ln a) a^x</math>

Thus ''any'' exponential function is a [[constant]] multiple of its own derivative.

If a variable's growth or decay rate is [[proportionality|proportional]] to its size, &mdash; as is the case in unlimited population growth (see [[Malthusian catastrophe]]), continuously compounded [[interest]], or [[radioactive decay,]] &mdash; then the variable can be written as a constant times an exponential function of time.

==Formal definition==

The exponential function thus solves the basic [[differential equation]]

: <math>{dy \over dx} = y</math>

and it is for this reason commonly encountered in differential equations. In particular the solution of linear ordinary [[differential equation]]s can frequently be written in terms of exponential functions. These equations include [[Schrödinger equation]] and the [[Laplace's equation]] as well as the equations for [[simple harmonic motion]].

ItThe exponential function e^''x'' can be defined in two equivalent ways:, as an [[infinite series]] or a [[limit]] of a sequence:

== Exponential function on the complex plane ==

: <math>e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots</math>

or as the [[limit of a sequence]]:

: <math>e^x = \lim_{n \to \infty} \left( 1 + {x \over n} \right)^n</math>

In these definitions, <math>n!</math> stands for the [[factorial]] of ''n'' and ''x'' can be any [[real number]], [[complex number]], element of a [[Banach algebra]] (for example, a [[square matrix]]), or member of the field of [[p-adic numbers|''p''-adic numbers]].

ToFor seefurther the equivalenceexplanation of these definitions and a proof of their equivalence, see the article [[Definitions of the exponential function]].

== Exponential function onOn the complex plane ==

When considered as a function defined on the [[complex number|complex plane]], the exponential function retains the important properties

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: <math>e^z \ne 0</math>

: <math>{d \over dz} e^z = e^z</math>

for all ''z'' and ''w''.

for all <i>z</i> and <i>w</i>. The exponential function on the complex plane is a [[holomorphic function]] which is periodic with imaginary period <math>2 \pi i</math> which can be written as

for all <i>z</i> and <i>w</i>. The exponential function on the complex planeIt is a [[holomorphic function]] which is periodic with [[imaginary number|imaginary]] period <math>2 \pi i</math> whichand can be written as

: <math>e^{a + bi} = e^a (\cos b + i \sin b)</math>

Thewhere exponential''a'' functionand also''b'' givesare risereal tovalues. theThis [[trigonometricformula function]]sconnects (asthe canexponential befunction seenwith fromthe [[Eulerstrigonometric formula in complex analysis|Euler's formulafunction]])s and to the [[hyperbolic function]]s. Thus we see that all [[elementary functionsfunction]]s except for the [[polynomial]]s spring from the exponential function in one way or another.

See also [[Eulers formula in complex analysis]] [[Euler's formula]].

where <i>a</i> and <i>b</i> are real values. This formula connects the exponential function with the [[Trigonometric function|trigonometric functions]], and this is the reason that extending the natural logarithm to complex arguments yields a [[multi-valued function]] ln(<i>z</i>). We can define a more general exponentiation:

Extending the natural logarithm to complex arguments yields a [[multi-valued function]], ln(''z''). We can then define a more general exponentiation:

: <math>z^w = e^{w \ln z}</math>

for all complex numbers ''z'' and ''w''. This is then also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.

for all complex numbers <i>z</i> and <i>w</i>.

This is then also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.

It is easy to see, that theThe exponential function maps any [[line]] in the complex plane to a [[logarithmic spiral]] in the complex plane with the centrecenter at 0,the [[origin]]. This can be seen by noting that the case of a line parallel with the real or imaginary axis maps to a line or [[circle]].

== Exponential function for matricesMatrices and Banach algebras ==

The definition of the exponential function exp given above can be used verbatim for every [[Banach algebra]], and in particular for square [[matrix_matrix (mathematics)|matrices]]. In this case we have

: <math>e^{x + y} = e^x e^y \mbox{ if } xy = yx</math>

if <math>xy = yx</math> (''we should add the general formula involving commutators here.'')

: <math>e^0 = 1</math>

:<i> ''e''^''x''</i> is invertible with inverse <i>''e</i>''^{-<i>&minus;''x</i>''}

:In addition, the derivative of exp at the point <i>''x</i>'' is that linear map which sends <i>''u</i>'' to <i>''u</i>''&nbsp;&middot;<i>&nbsp;''e''^''x''</i>.

In the context of non-commutative Banach algebras, such as algebras of matrices or operators on [[Banach space|Banach]] or [[Hilbert space|Hilbert]] spaces, the exponential function is often considered as a function of a real argument:

: <math>f(t) = e^{t A}</math>

where <math>''A</math>'' is a fixed element of the algebra and <math>''t</math>'' is any real number. This function has the important properties

: <math>f(s + t) = f(s) f(t)</math>

: <math>f(0) = 1</math>

: <math>f'(t) = A f(t)</math>

== Exponential map onOn Lie algebras ==

The "exponential map" sending a [[Lie algebra]] to the [[Lie group]] that gave rise to it shares the above properties, which explains the terminology. In fact, since '''R''' is the Lie algebra of the Lie group of all positive real numbers with multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie algebra M(<i>''n</i>'', '''R''') of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.

The "exponential map" sending a [[Lie algebra]] to the [[Lie group]] that gave rise to it shares the above properties, which explains the terminology. In fact, since '''R''' is the Lie algebra of the Lie group of all positive real numbers with multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie algebra M(<i>n</i>, '''R''') of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.

==See also==

*[[exponential growth]]

[[exponential growth]]