Exponential function: Difference between revisions

Revision as of 07:34, 29 February 2004

Revision as of 00:38, 18 March 2004

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[[de:Exponentialfunktion]]

[[fr:Exponentielle]][[pl:Funkcja_wyk%C5%82adnicza]]

The '''exponential function''' is one of the most important [[function]]s in [[mathematics]]. It is written as exp(''<i>x''</i>) or <mathi>e^<sup>x</mathsup></i> (where ''<i>e''</i> is the [[e (mathematical constant)|base of the natural logarithm]]) and can be defined in two equivalent ways, the first an [[infinite series]], the second a [[limit]]:

<div style="float:right;width:201px;margin-left:0.5em;">

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

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

</div>

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

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

Here <math>n!</math> stands for the [[factorial]] of <mathi>n</mathi> and <mathi>x</mathi> 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 ''<i>x''</i> is real, then exp(''<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 as follows:

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

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

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.

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

: <math>a^{x + y} = a^x a^y</math>

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

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

: <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>. Expressions involving fractions and roots can often be simplified using exponential notation because :

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

: <math>\sqrt{a} = a^{1/2}</math>

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=== Exponential function and differential equations ===

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

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

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.

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

: <math>\exp(e^{z + w)} = \exp(e^z) \exp(e^w)</math>

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

: <math>\exp(e^z) \ne 0</math>

: <math>{d \exp'(over dz} e^z) = \exp(e^z)</math>

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

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

where <mathi>a</mathi> and <mathi>b</mathi> are real values. This formula connects the exponential function with the [[trigonometricTrigonometric function|trigonometric functions]]s, 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:

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

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.

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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_(mathematics)|matrices]]. In this case we have

: <math>\exp(e^{x + y)} = \exp(e^x) \exp(e^y)</math>

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

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

:exp(''<i>e^x'')</i> is invertible with inverse exp(<i>e</i>^{-''<i>x'')</i>}

:the derivative of exp at the point ''<i>x''</i> is that linear map which sends ''<i>u''</i> to exp(''x'')<i>u</i>&middot;''u''<i>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) = \exp(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>

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=== Exponential map on 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.

See also [[exponential growth]].