Editing Functional equation

{{Short description|Equation whose unknown is a function}}
{{distinguish|Functional model}}

In [[mathematics]], a '''functional equation'''
<ref name="rassias">{{cite book | title=Functional Equations and Inequalities | last=Rassias | first=Themistocles M. | year=2000 | publisher=[[Kluwer Academic Publishers]] | location=3300 AA Dordrecht, The Netherlands | isbn=0-7923-6484-8 | page= 335 | url=https://books.google.com/books?id=tFTFBAAAQBAJ&q=%22Introduction+to+the+Theory+of+Functional+Equations+and+Inequalities%22 }}</ref><ref name="rassias4"> {{cite book |title=Functional Equations and Inequalities in Several Variables |last=Czerwik |first=Stephan |year=2002 |publisher=[[World Scientific Publishing Co.]] |location=P O Box 128, Farrer Road, Singapore 912805 |isbn=981-02-4837-7 |page= [https://archive.org/details/functionalequati00czer_083/page/n419 410] |url=https://archive.org/details/functionalequati00czer_083 |url-access=limited }}</ref>{{irrelevant citation|reason=What fact or sentence is being cited here? These citations don't make any sense.|date=March 2022}} is, in the broadest meaning, an [[equation]] in which one or several functions appear as [[unknown (mathematics)|unknown]]s. So, [[differential equation]]s and [[integral equation]]s are functional equations. However, a more restricted meaning is often used, where a ''functional equation'' is an equation that relates several values of the same function. For example, the [[logarithm function]]s are [[logarithm#Characterization by the product formula|essentially characterized]] by the ''logarithmic functional equation'' <math>\log(xy)=\log(x) + \log(y).</math>

If the [[domain of a function|domain]] of the unknown function is supposed to be the [[natural number]]s, the function is generally viewed as a [[sequence (mathematics)|sequence]], and, in this case, a functional equation (in the narrower meaning) is called a [[recurrence relation]]. Thus the term ''functional equation'' is used mainly for [[real function]]s and [[complex function]]s. Moreover a [[smooth function|smoothness condition]] is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the  [[gamma function]] is a function that satisfies the functional equation <math>f (x + 1) = x f (x)</math> and the initial value <math>f (1) = 1.</math> There are many functions that satisfy these conditions, but the gamma function is the unique one that is [[meromorphic function|meromorphic]] in the whole complex plane, and [[logarithmically convex function|logarithmically convex]] for {{mvar|x}} real and positive ([[Bohr–Mollerup theorem]]).

==Examples==


*[[Recurrence relation]]s can be seen as functional equations in functions over the integers or natural numbers, in which the differences between terms' indexes can be seen as an application of the [[shift operator]]. For example, the recurrence relation defining the [[Fibonacci numbers]], <math>F_{n} = F_{n-1}+F_{n-2}</math>, where <math>F_0=0</math> and <math>F_1=1</math>

*<math>f(x+P) = f(x)</math>, which characterizes the [[periodic function]]s

*<math>f(x) = f(-x)</math>, which characterizes the [[even function]]s, and likewise <math>f(x) = -f(-x)</math>, which characterizes the [[odd function]]s

*<math>f(f(x)) = g(x)</math>, which characterizes the [[functional square root]]s of the function g

*<math>f(x + y) = f(x) + f(y)\,\!</math> ([[Cauchy's functional equation]]), satisfied by [[linear map]]s. The equation may, contingent on the [[axiom of choice]], also have other pathological nonlinear solutions, whose existence can be proven with a [[Hamel basis]] for the real numbers
*<math>f(x + y) = f(x)f(y), \,\!</math> satisfied by all [[exponential function]]s. Like Cauchy's additive functional equation, this too may have pathological, discontinuous solutions
*<math>f(xy) = f(x) + f(y)\,\!</math>, satisfied by all [[logarithm]]ic functions and, over coprime integer arguments, [[additive function]]s
*<math>f(xy) = f(x) f(y)\,\!</math>, satisfied by all [[power function]]s and, over coprime integer arguments, [[multiplicative function]]s
*<math>f(x + y) + f(x - y) = 2[f(x) + f(y)]\,\!</math> (quadratic equation or [[parallelogram law]])
*<math>f((x + y)/2) = (f(x) + f(y))/2\,\!</math> ([[Jensen's functional equation]])
*<math>g(x + y) + g(x - y) = 2[g(x) g(y)]\,\!</math> ([[d'Alembert's functional equation]])
*<math>f(h(x)) = h(x + 1)\,\!</math> ([[Abel equation]])
*<math>f(h(x)) = cf(x)\,\!</math> ([[Schröder's equation]]).
*<math>f(h(x)) = (f(x))^c\,\!</math> ([[Böttcher's equation]]).
*<math>f(h(x)) = h'(x)f(x)\,\!</math> ([[Schröder's equation#Functional significance|Julia's equation]]).
*<math>f(xy) = \sum g_l(x) h_l(y)\,\!</math> (Levi-Civita),
*<math>f(x+y) = f(x)g(y)+f(y)g(x)\,\!</math> ([[List of trigonometric identities#Angle sum and difference identities|sine addition formula]] and [[Hyperbolic functions|hyperbolic sine addition formula]]),
*<math>g(x+y) = g(x)g(y)-f(y)f(x)\,\!</math> ([[List of trigonometric identities#Angle sum and difference identities|cosine addition formula]]),
*<math>g(x+y) = g(x)g(y)+f(y)f(x)\,\!</math> ([[Hyperbolic functions|hyperbolic cosine addition formula]]).
*The [[commutative law|commutative]] and [[associative law]]s are functional equations. In its familiar form, the associative law is expressed by writing the [[binary operation]] in [[infix notation]], <math display="block">(a \circ b) \circ c = a \circ (b \circ c)~,</math> but if we write ''f''(''a'',&thinsp;''b'') instead of {{math|''a'' ○ ''b''}} then the associative law looks more like a conventional functional equation, <math display="block">f(f(a, b),c) = f(a, f(b, c)).\,\!</math>

* The functional equation <math display="block">
f(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)f(1-s)
</math> is satisfied by the [[Riemann zeta function]], as proved [[Riemann_zeta_function#Riemann's_functional_equation|here]]. The capital {{math|Γ}} denotes the [[gamma function]].

* The gamma function is the unique solution of the following system of three equations:{{cn|date=March 2022}}
**<math>f(x)={f(x+1) \over x}</math>
**<math>f(y)f\left(y+\frac{1}{2}\right)=\frac{\sqrt{\pi}}{2^{2y-1}}f(2y)</math>
**<math>f(z)f(1-z)={\pi \over \sin(\pi z)}</math>{{spaces|10}}([[Leonhard Euler|Euler's]] [[reflection formula]])
* The functional equation <math display="block">f\left({az+b\over cz+d}\right) = (cz+d)^k f(z)</math> where {{math|''a'', ''b'', ''c'', ''d''}} are [[integer]]s satisfying <math>ad - bc = 1</math>, i.e. <math>
\begin{vmatrix} a & b\\ c & d \end{vmatrix}</math> = 1, defines {{mvar|f}} to be a [[modular form]] of order {{mvar|k}}.

One feature that all of the examples listed above have in common is that, in each case, two or more known functions (sometimes multiplication by a constant, sometimes addition of two variables, sometimes the [[identity function]]) are inside the argument of the unknown functions to be solved for.

When it comes to asking for ''all'' solutions, it may be the case that conditions from [[mathematical analysis]] should be applied; for example, in the case of the ''Cauchy equation'' mentioned above, the solutions that are [[continuous function]]s are the 'reasonable' ones, while other solutions that are not likely to have practical application can be constructed (by using a [[Hamel basis]] for the [[real number]]s as [[vector space]] over the [[rational number]]s). The [[Bohr–Mollerup theorem]] is another well-known example.

===Involutions===

The [[involution (mathematics)|involution]]s are characterized by the functional equation <math>f(f(x)) = x</math>. These appear in [[Charles Babbage|Babbage's]] functional equation (1820),<ref>{{Cite journal | doi = 10.2307/2007270| jstor = 2007270| title = On Certain Real Solutions of Babbage's Functional Equation| journal = The Annals of Mathematics| volume = 17| issue = 3| pages = 113–122| year = 1916| last1 = Ritt | first1 = J. F. | author1-link = Joseph Ritt}}</ref>
: <math>f(f(x)) = 1-(1-x) = x \, .</math>

Other involutions, and solutions of the equation, include

*<math> f(x) = a-x\, ,</math>
*<math> f(x) = \frac{a}{x}\, ,</math> and 
*<math> f(x) = \frac{b-x}{1+cx} ~ ,</math>
which includes the previous three as [[special case]]s or limits.

==Solution==
One method of solving elementary functional equations is substitution.{{citation needed|date=March 2022}}

Some solutions to functional equations have exploited [[surjective|surjectivity]], [[injective function|injectivity]], [[odd function|oddness]], and [[even function|evenness]].{{citation needed|date=March 2022}}

Some functional equations have been solved with the use of [[ansatz]]es, [[mathematical induction]].{{citation needed|date=March 2022}}

Some classes of functional equations can be solved by computer-assisted techniques.{{vague|reason="Computer assisted" is far too vague. What kinds of techniques were these?|date=March 2022}}<ref>{{Cite journal|last=Házy|first=Attila| date=2004-03-01| title=Solving linear two variable functional equations with computer|journal=Aequationes Mathematicae| language=en| volume=67|issue=1|pages=47–62|doi=10.1007/s00010-003-2703-9|s2cid=118563768|issn=1420-8903}}</ref>

In [[dynamic programming]] a variety of successive approximation methods<ref>Bellman, R. (1957). Dynamic Programming, [[Princeton University Press]].</ref><ref>Sniedovich, M. (2010). Dynamic Programming: Foundations and Principles, [[Taylor & Francis]].</ref> are used to solve [[Bellman equation|Bellman's functional equation]], including methods based on [[fixed point iteration]]s.

==See also==
*[[Functional equation (L-function)]]
*[[Bellman equation]]
*[[Dynamic programming]]
*[[Implicit function]]
*[[Functional differential equation]]

==Notes==
<references/>

==References==
*[[János Aczél (mathematician)|János Aczél]], ''[https://books.google.com/books?id=JEB0BFvRwrcC Lectures on Functional Equations and Their Applications]'', [[Academic Press]], 1966, reprinted by Dover Publications, {{ISBN|0486445232}}.
*János Aczél & J. Dhombres, ''[https://books.google.com/books?id=8EWnEh18rVgC&q=%22Functional+Equations+in+Several+Variables%22 Functional Equations in Several Variables]'', [[Cambridge University Press]], 1989.
*C. Efthimiou, ''Introduction to Functional Equations'', AMS, 2011, {{isbn|978-0-8218-5314-6}} ; [https://web.archive.org/web/20230603025341/http://www.msri.org/people/staff/levy/files/MCL/Efthimiou/100914book.pdf online].
*Pl. Kannappan, ''[https://books.google.com/books?id=SdZoCM2OeuIC&dq=%22Functional+Equations+and+Inequalities+with+Applications%22&pg=PA1 Functional Equations and Inequalities with Applications]'', Springer, 2009.
*[[Marek Kuczma]], ''[https://books.google.com/books?id=tFTFBAAAQBAJ&q=%22Introduction+to+the+Theory+of+Functional+Equations+and+Inequalities%22 Introduction to the Theory of Functional Equations and Inequalities]'', second edition, Birkhäuser, 2009.
*Henrik Stetkær, ''[https://books.google.com/books?id=JzS7CgAAQBAJ&dq=%22Functional+Equations+on+Groups%22&pg=PR5 Functional Equations on Groups]'', first edition, World Scientific Publishing, 2013.
*{{cite book|author=Christopher G. Small|title=Functional Equations and How to Solve Them|url=https://books.google.com/books?id=2D2RYbb22nMC|date=3 April 2007|publisher=Springer Science & Business Media|isbn=978-0-387-48901-8}}

== External links ==
* [http://eqworld.ipmnet.ru/en/solutions/fe.htm Functional Equations: Exact Solutions] at EqWorld: The World of Mathematical Equations.
* [http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-fe.htm Functional Equations: Index] at EqWorld: The World of Mathematical Equations.
* [https://web.archive.org/web/20120227145129/http://www.imomath.com/tekstkut/funeqn_mr.pdf IMO Compendium text (archived)] on functional equations in problem solving.

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[[Category:Functional equations| ]]