Exponential function: Difference between revisions

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

[[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]]) and can be defined in either of two equivalent ways, the first an [[infinite series]], the second a [[limit]] of a sequence:.

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

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

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.

[[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.

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

: <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>

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

: <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>

Here <math>n!</math> stands for the [[factorial]] of <i>n</i> and <i>x</i> 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]].

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

Here, <math>n!</math> stands for the [[factorial]] of <i>n</i> and <i>x</i> 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]].

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 as follows:

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

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

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.

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: <math>\sqrt[c]{a}^b = a^{b \over c}</math>

== 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 [[Derivative|derivatives]]: