Content deleted Content added
Tito Omburo (talk | contribs) log(z) |
Tito Omburo (talk | contribs) |
||
| (29 intermediate revisions by 6 users not shown) | |||
Line 7:
In an attempt to solve this problem, [[Joseph Liouville]] introduced in 1833 a definition of elementary functions that extends the above one and is commonly accepted:<ref>{{harvnb|Liouville|1833a}}.</ref><ref>{{harvnb|Liouville|1833b}}.</ref><ref>{{harvnb|Liouville|1833c}}.</ref> An ''elementary function'' is a function that can be built, using addition, multiplication, division, and function composition, from [[constant function]]s, exponential functions, the [[complex logarithm]], and [[polynomial roots|roots]] of polynomials with elementary functions as coefficients. This includes the trigonometric functions, since, for example, {{tmath|1=\textstyle \cos x=\frac{e^{ix}+e^{-ix} }2}}, as well as every [[algebraic function]].
[[Liouville's theorem (differential algebra)|Liouville's result]] is that, if an elementary function has an elementary antiderivative, then this antiderivative is a linear combination of logarithms, where the coefficients and the arguments of the logarithms are elementary functions involved, in some sense, in the definition of the function. More than 130 years later, [[Risch algorithm]], named after [[Robert Henry Risch]], is an algorithm to decide whether an elementary function has an elementary antiderivative, and, if it has, to compute this antiderivative. Despite dealing with elementary functions, the Risch algorithm is far from elementary; {{as of|2025|lc=y}}, it seems that no complete implementation is available.
== Examples ==
Line 13:
=== Basic examples ===
Elementary functions of a single variable {{mvar|x}} include:
* [[Constant function]]s: <math>2,\ \pi,\ e,</math>
* [[Exponentiation|Powers of {{
* [[Exponential function]]s: <math>\textstyle e^x,
* [[Logarithm]]s: <math>\textstyle \log x, \
* [[Trigonometric function]]s: <math>\textstyle\sin x=\frac{e^{ix}-e^{-ix}}{2i},\ \cos x=\frac{e^{ix}+e^{-ix}}{2},\ \tan x=\frac{\sin x}{\cos x},\ </math> etc.
* [[Inverse trigonometric function]]s: <math>\arcsin x,\ \arccos x,</math> etc.
* [[Hyperbolic function]]s: <math>\sinh x,\ \cosh x,</math> etc.
* [[Inverse hyperbolic function]]s: <math>\operatorname{arsinh} x,\ \operatorname{arcosh} x,</math> etc.
* All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions<ref>{{cite book|author=Morris Tenenbaum|title=Ordinary Differential Equations|date=1985|publisher=Dover|isbn=0-486-64940-7|page=[https://archive.org/details/ordinarydifferen00tene_0/page/17 17]|url-access=registration|url=https://archive.org/details/ordinarydifferen00tene_0/page/17}}</ref>
* All functions obtained
* All functions obtained by [[function composition|composing]] a finite number of any of the previously listed functions
Line 41:
=== Non-elementary functions ===
All elementary functions are [[Analytic function|analytic]]
Not every analytic function is elementary.
* the [[gamma function]]
* non-elementary [[Liouvillian function#Examples|Liouvillian functions]], including
** the [[exponential integral]] (''Ei'')
** the [[error function]], <math>\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt,</math> a fact that may not be immediately obvious, but can be proven using the [[Risch algorithm]]
* other [[nonelementary integral]]s, including the [[Dirichlet integral]] and [[elliptic integral]]
== Closure ==
Line 56 ⟶ 54:
==Differential algebra==
Some have proposed extending the set of elementary functions by extending with certain [[transcendental function]]s, to include, for example, the [[Lambert W function]]<ref>{{Cite journal |last=Stewart |first=Seán |date=2005 |title=A new elementary function for our curricula? |url=https://files.eric.ed.gov/fulltext/EJ720055.pdf |journal=Australian Senior Mathematics Journal |volume=19 |issue=2 |pages=8–26}}</ref> or [[elliptic function]]s,<ref>Ince, E. L. (1956) [1926]. ''Ordinary Differential Equations''. New York: Dover Publications. ISBN 0-486-60339-4, footnote to p 330</ref> all of which are analytic. The key attribute, from the perspective of the Liouville theorem, is that as a class, they are closed under taking derivatives. For example, the Lambert function <math>w=W(z)</math>, which is defined implicitly by the equation <math>we^w=z</math>, has a derivative which can be obtained by [[implicit differentiation]]:
<math>W'(z) = \frac{e^{-W(z)}}{1+W(z)},</math>
which is again "elementary", provided that <math>W(z)</math> is.
The mathematical definition of an
A ''differential field'' {{tmath|F}} is a field together with a [[derivation (differential algebra)|derivation]] {{tmath|u\mapsto \partial u}} that maps {{tmath|F}} to itself. The derivation generalizes [[derivative]], being linear (thaat is, {{tmath|1=\partial (u + v) = \partial u + \partial v}}) and satisfying the [[product rule|Leibniz product rule]] (that is,{{tmath|1=\partial(u\cdot v)=\partial u\cdot v+u\cdot\partial v}}) for every two elements {{tmath|u}} and {{tmath|v}} in {{tmath|F}}. The [[rational function]]s over {{tmath|\Q}} of {{tmath|\C}} form a basic examples of differential fields, when equipped with the usual derivative.
An element {{math|''h''}} of {{tmath|F}} is a constant if {{tmath|1=\partial h=0}}. The constants of {{tmath|F}} form a dfferential field with zero derivative. Care must be taken that a differential field extension of a differential field may enlarge the field of constants.
A function {{mvar|u}} of a differential extension {{mvar|G}} of a differential field {{mvar|F}} is an '''elementary function''' over {{mvar|F}} if it belongs to a finite chain (for inclusion) of differential subfields of {{mvar|G}} that starts from {{mvar|F}} and is such that each is generated over the preceding one by a function that is either
*
* a ''logarithm'', that is, {{tmath|1=\partial u = \partial a/a}} for some {{tmath|a\in F}}.
With this definition, the usual elementary functions are exactly the function that are elementary over the field of the [[rational function]]s. This generalized definition allows considering every transcendental function as elementary for applying Liouville's theorem.
▲* is [[Algebraic function|algebraic]] over ''F'', or
▲* is an '''exponential''', that is, ∂''u'' = ''u'' ∂''a'' for ''a'' ∈ ''F'', or
▲(see also [[Liouville's theorem (differential algebra)|Liouville's theorem]])
==See also==
| |||