Exponential function: Difference between revisions

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The '''exponential function''' is one of the most important [[function|functions]] in [[mathematics]]. It is written as exp(''x'') or ''e''<supmath>''e^x''</supmath> (where ''<math>e''</math> is the [[e - base of natural logarithm|base of the natural logarithm]]) and can be defined in two equivalent ways:

: <math>\exp(x) = \sum_{n = 0}^{\infty} {x^n \over n!}</math>

&infin; ''x''^''n''

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

exp(''x'') = &sum; ---

(see [[limit]] and [[infinite series]]). Here ''<math>n''!</math> stands for the [[factorial]] of ''<math>n''</math> and ''<math>x''</math> can be any [[real number|real]] or [[complex number|complex]] number, or even any element of a [[Banach algebra]] or the field of [[p-adic numbers|<i>p</i>-adic numbers]].

''n''=0 ''n''!

exp(''x'') = lim (1 + ''x''/''n'')^''n''

''n''&rarr;&infin;

(see [[limit]] and [[infinite series]]). Here ''n''! stands for the [[factorial]] of ''n'' and ''x'' can be any [[real number|real]] or [[complex number|complex]] number, or even any element of a [[Banach algebra]] or the field of [[p-adic numbers|<i>p</i>-adic numbers]].

If ''x'' is real, then exp(''x'') is positive and strictly increasing. Therefore its [[inverse function]], the [[natural logarithm]] ln(''x''), is defined for all positive ''x''. Using the natural logarithm, one can define more general exponential functions as follows:

: <math>a^x = \exp(\ln(a) x)</math>

for all ''<math>a'' > 0</math> and all real ''<math>x''</math>.

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

:''a''^''x'' = exp(ln(''a'') ''x'')

for all ''a'' > 0 and all real ''x''.

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

:''a'' <supmath>a^0</sup> = 1</math>

:''a'' <supmath>a^1</sup> = ''a''</math>

:''a'' <supmath>''a^{x'' + ''y''</sup> } = ''a''^{''^x''} ''a''<sup>''^y''</supmath>

: <math>a^{x y} = \left( a^x \right)^y</math>

:''a''^(''xy'') = (''a''^''x'')^''y''

: <math>{1 /\over ''a''^{''^x''} } = \left (1/'' \over a'' \right)^{''^x''} = ''a''<sup>^{-''x''}</supmath>

:''a'' <supmath>''a^x''</sup> ''b''^{''^x''} = (a (''ab''b)<sup>''^x''</supmath>

These are valid for all positive real numbers ''a'' and ''b'' and all real numbers ''x''. Expressions involving fractions and roots can often be simplified using exponential notation because

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

:&radic; ''<math>\sqrt{a'' } = ''a''<sup>^{1/2}</supmath>

: <supmath>''\sqrt[n''</sup>&radic; '']{a'' } = ''a''<sup>^{1/''n''}</supmath>

=== Exponential function and differential equations ===

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The major importance of the exponential functions in the sciences stems from the fact that they are constant multiples of their own [[derivative|derivatives]]:

: <math>{d/d''x'' ''\over dx} a''^{''^{bx''} } = \ln(''a'') ''b'' ''a''<sup>''^{bx''}.</supmath>.

If a variable's growth or decay rate is proportional to its size, as is the case in unlimited population growth, continuously compounded interest or radioactive decay, then the variable can be written as a constant times an exponential function of time.

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 [[Schrodinger's equation]] and the [[Laplace equation]] as well as the equations for [[simple harmonic motion]].

dy

-- = y

dx

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 [[Schrodinger's equation]] and the

[[Laplace equation]] as well as the equations for [[simple harmonic motion]].

=== Exponential function on the complex plane ===

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

: <math>\exp(z + w) = \exp(z) \exp(w)</math>

: <math>\exp(0) = 1</math>

: <math>\exp(z) \neq 0</math>

: <math>\exp'(''z'') = \exp(''z'')

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

: <math>\exp(''z''a + ''w''bi) = \exp(''z''a) exp\cdot (''w''\cos(b) + i * \sin(b))</math>

:exp(0) = 1

:exp(''z'') &ne; 0

:exp'(''z'') = exp(''z'')

for all ''z'' and ''w''. The exponential function on the complex plane is a [[holomorphic function]] which

is periodic with imaginary period 2&pi;''i'' which can

be written as

:exp(''a'' + ''bi'') = exp(''a'') * (cos(''b'') + i * sin(''b''))

where ''a'' and ''b'' are real values. This formula

connects the exponential function with the [[trigonometric function]]s, and this is the reason that extending the natural logarithm to complex arguments naturally yields a multi-valued function ln(''z''). We can define a more general exponentiation:

:''z''^''w'' = exp(ln(''z'') ''w'')

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

: <math>z^w = \exp(\ln(z) w)</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.

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

: <math>\exp(''x'' + ''y'') = \exp(''x'') \exp(''y'') &nbsp;&nbsp;&nbsp; </math>
if ''<math>xy'' = ''yx''</math> (''we should add the general formula involving commutators here.'')

: <math>\exp(0) = 1</math>

:exp(''x'') is invertible with inverse exp(-''x'')

:the derivative of exp at the point ''x'' is that linear map which sends ''u'' to exp(''x'')&middot;''u''.

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) = \exp(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

:''f''(''t'') = exp(''t'' ''A'')

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

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

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

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

:''f''(''s'' + ''t'') = ''f''(''s'') ''f''(''t'')

:''f''(0) = 1

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

=== Exponential map on Lie algebras ===