Functional equation: Difference between revisions
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imported>Dmitrii Kouznetsov (add references. Need to cite several monographiies ant textbooks. Oops.. So many articles are still missed...) |
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In [[mathematics]], a '''functional equation''' is | In [[mathematics]], a '''functional equation''' is an implicit way to specify some [[mathematical function]] | ||
<ref name="cheng"> | |||
{{cite book | |||
|title=Analytic solutions of Functional equations | |||
|last=Cheng | |||
|first=Sui Sun | |||
|authorlink= | |||
|coauthors=Wendrong Li | |||
|year=2008 | |||
|publisher=[[World Scientific Publishing Co.]] | |||
|location=5 Toh Tuck Link, Singapore 596224 | |||
|isbn=13 978-981-279-334-8 | |||
|page= | |||
|pages= | |||
|url= | |||
|accessdate=}} | |||
</ref>. | |||
Often, the functional equation relatte values of a [[function (mathematics)|function]] at different arguments. Usually, the solution of a functional equation is supposed to be a [[holomorphic function]], although some functional equations were initially established for functions of a discrete variable; see for example the [[Ackermann functions]]. | |||
==Examples== | ==Examples== | ||
* Any [[periodic function]] has a functional equation of the form <math>f(x+p) = f(x)</math> where ''p'' is the period. | * Any [[periodic function]] has a functional equation of the form <math>f(x+p) = f(x)</math> where ''p'' is the period. | ||
| Line 7: | Line 23: | ||
* The [[gamma function]] has a functional equation relating <math>\Gamma(z)</math> to <math>\Gamma(z-1)</math>. | * The [[gamma function]] has a functional equation relating <math>\Gamma(z)</math> to <math>\Gamma(z-1)</math>. | ||
* The [[Riemann zeta function]] has a function equation relating the value of <math>\zeta(s)</math> to <math>\zeta(1-s)</math>. | * The [[Riemann zeta function]] has a function equation relating the value of <math>\zeta(s)</math> to <math>\zeta(1-s)</math>. | ||
* [[Superfunction]] <math>F</math> of a given function <math>H</math> is solution of functional equation <math>H(F(z))=F(z+1)</math>. | |||
* [[Abel equation]] | |||
* [[Schroeder equation]] | |||
==See also== | |||
*[[Linear equation]] | |||
*[[Algebraic equation]] | |||
*[[Differential equation]] | |||
*[[Functional analysis]] | |||
==References== | |||
<references/> | |||
Revision as of 23:05, 8 September 2009
In mathematics, a functional equation is an implicit way to specify some mathematical function [1]. Often, the functional equation relatte values of a function at different arguments. Usually, the solution of a functional equation is supposed to be a holomorphic function, although some functional equations were initially established for functions of a discrete variable; see for example the Ackermann functions.
Examples
- Any periodic function has a functional equation of the form where p is the period.
- Examples of periodic functions include trigonometric functions and elliptic functions.
- The gamma function has a functional equation relating to .
- The Riemann zeta function has a function equation relating the value of to .
- Superfunction of a given function is solution of functional equation .
- Abel equation
- Schroeder equation
See also
References
- ↑ Cheng, Sui Sun; Wendrong Li (2008). Analytic solutions of Functional equations. 5 Toh Tuck Link, Singapore 596224: World Scientific Publishing Co.. ISBN 13 978-981-279-334-8.