Exponential function: Difference between revisions

Revision as of 12:33, 19 June 2003

Revision as of 00:32, 29 June 2003

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[[de:Exponentialfunktion]]

[[fr:Exponentielle]]

The '''exponential function''' is one of the most important [[function|functions]]s in [[mathematics]]. It is written as exp(''x'') or <math>e^x</math> (where ''e'' is the [[e - base of natural logarithm|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:201;margin-left:0.5em;">

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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 [[derivative|derivatives]]s:

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

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

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