Editing Elementary function (section)

==Differential algebra==

The mathematical definition of an '''elementary function''', or a function in elementary form, is considered in the context of [[differential algebra]].  A differential algebra is an algebra with the extra operation of derivation (algebraic version of differentiation).  Using the derivation operation new equations can be written and their solutions used in [[field extension|extensions]] of the algebra.  By starting with the [[field (mathematics)|field]] of [[rational function]]s, two special types of transcendental extensions (the logarithm and the exponential) can be added to the field building a tower containing elementary functions.

A '''differential field''' ''F'' is a field ''F''<sub>0</sub> (rational functions over the [[rational number|rationals]] '''Q''' for example) together with a derivation map ''u''&nbsp;→&nbsp;∂''u''.  (Here ∂''u'' is a new function. Sometimes the notation ''u''&prime; is used.) The derivation captures the properties of differentiation, so that for any two elements of the base field, the derivation is linear

: <math>\partial (u + v) = \partial u + \partial v </math>

and satisfies the [[product rule|Leibniz product rule]]

: <math>\partial(u\cdot v)=\partial u\cdot v+u\cdot\partial v\,.</math>

An element ''h''  is a constant if ''∂h&nbsp;=&nbsp;0''.   If the base field is over the rationals, care must be taken when extending the field to add the needed transcendental constants.

A function ''u'' of a differential extension ''F''[''u''] of a differential field ''F'' is an '''elementary function''' over ''F'' if the function ''u''
* is [[Algebraic function|algebraic]] over ''F'', or
* is an '''exponential''', that is, ∂''u'' = ''u'' ∂''a''  for ''a'' ∈ ''F'', or
* is a '''logarithm''', that is, ∂''u'' = ∂''a''&nbsp;/&nbsp;a for ''a'' ∈ ''F''.
(see also [[Liouville's theorem (differential algebra)|Liouville's theorem]])