|
|
| Line 1: |
Line 1: |
| | | My name's Carla Marroquin but everybody calls me Carla. I'm from France. I'm studying at the high school (2nd year) and I play the Piano for 5 years. Usually I choose songs from my famous films :D. <br>I have two sister. I like Color Guard, watching TV (Psych) and Machining.<br><br>my homepage: [http://perbiz.co.kr/xe/?document_srl=2524931 Hostgator 1 cent coupon] |
| In [[mathematics]], a '''functional equation'''<ref name="rassias">
| |
| {{cite book
| |
| |title=Functional Equations and Inequalities
| |
| |last=Rassias
| |
| |first=Themistocles M.
| |
| |authorlink=
| |
| |coauthors=
| |
| |year=2000
| |
| |publisher=[[Kluwer Academic Publishers]]
| |
| |location=3300 AA Dordrecht, The Netherlands
| |
| |isbn=0- 7923-6484-8
| |
| |page= 335
| |
| |url=
| |
| |accessdate=}}
| |
| </ref><ref name="rassias2">
| |
| {{cite book
| |
| |title=Stability of Functional Equations in Several Variables
| |
| |last= Hyers, D. H.
| |
| |first=
| |
| |authorlink=
| |
| |coauthors= Isac, G., Rassias, Th. M.
| |
| |year=1998
| |
| |publisher=[[Birkhäuser Verlag]]
| |
| |location=Boston
| |
| |isbn=0-8176-4024-X
| |
| |page= 313
| |
| |url=
| |
| |accessdate=}}
| |
| </ref><ref name="rassias3">
| |
| {{cite book
| |
| |title=Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis
| |
| |last=Jung
| |
| |first=Soon-Mo
| |
| |authorlink=
| |
| |coauthors=
| |
| |year=2001
| |
| |publisher=[[Hadronic Press, Inc.]]
| |
| |location=35246 US 19 North # 115, Palm Harbor, FL 34684 USA
| |
| |isbn=1-57485-051-2
| |
| |page= 256
| |
| |url=
| |
| |accessdate=}}
| |
| </ref><ref name="rassias4">
| |
| {{cite book
| |
| |title=Functional Equations and Inequalities in Several Variables
| |
| |last=Czerwik
| |
| |first=Stephan
| |
| |authorlink=
| |
| |coauthors=
| |
| |year=2002
| |
| |publisher=[[World Scientific Publishing Co.]]
| |
| |location=P O Box 128, Farrer Road, Singapore 912805
| |
| |isbn=981-02-4837-7
| |
| |page= 410
| |
| |url=
| |
| |accessdate=}}
| |
| </ref> is any equation that specifies a function in [[Implicit function|implicit form]].<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= 978-981-279-334-8
| |
| |page=
| |
| |pages=
| |
| |url=
| |
| |accessdate=}}
| |
| </ref>
| |
| Often, the [[equation]] relates the value of a function (or functions) at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional equations they satisfy. The term ''functional equation'' usually refers to equations that cannot be simply reduced to algebraic equations.
| |
| <!--, often because two or more known functions are substituted as arguments into another function.
| |
| !-->
| |
| | |
| ==Examples==
| |
| | |
| * The functional equation
| |
| | |
| ::<math>
| |
| 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]] ζ. The capital Γ denotes the [[gamma function]].
| |
| | |
| * These functional equations are satisfied by the gamma function. The gamma function is the unique solution of the system of all three equations:
| |
| ::<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> ([[Leonhard Euler|Euler's]] [[reflection formula]])
| |
| | |
| * The functional equation
| |
| | |
| ::<math>f\left({az+b\over cz+d}\right) = (cz+d)^k f(z)\,\!</math>
| |
| | |
| :where ''a'', ''b'', ''c'', ''d'' are [[integer]]s satisfying ''ad'' − ''bc'' = 1, i.e. <math>
| |
| \begin{vmatrix} a & b\\c & d\end{vmatrix}\,=1</math>, defines ''f'' to be a [[modular form]] of order ''k''.
| |
| | |
| * Miscellaneous examples not necessarily involving "famous" functions:
| |
| | |
| ::<math>f(x + y) = f(x)f(y), \,\!</math> satisfied by all [[exponential function]]s
| |
| | |
| ::<math>f(xy) = f(x) + f(y)\,\!</math>, satisfied by all [[logarithm]]ic functions
| |
| | |
| ::<math>f(x + y) = f(x) + f(y)\,\!</math> ([[Cauchy functional equation]])
| |
| | |
| ::<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)
| |
| | |
| ::<math>g(x + y) + g(x - y) = 2[g(x) g(y)]\,\!</math> (d'Alembert)
| |
| | |
| ::<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(x+y) = f(x)g(y)+f(y)g(x)\,\!</math> (sine addition formula).
| |
| | |
| ::<math>g(x+y) = g(x)g(y)-f(y)f(x)\,\!</math> (cosine addition formula).
| |
| | |
| ::<math>f(xy) = \sum g_l(x) h_l(y)\,\!</math> (Levi-Civita).
| |
| | |
| *A simple form of functional equation is a [[recurrence relation]]. This, formally speaking, involves an unknown function on integers, and also translation operators.
| |
| | |
| :One such example of a recurrence relation is
| |
| | |
| :: <math>a(n) = 3a(n-1) + 4a(n-2)\,\!</math>
| |
| *The commutative and associative laws are functional equations. When the associative law is expressed in its familiar form, one lets some symbol between two variables represent a binary operation, thus:
| |
| | |
| ::<math>(a*b)*c = a*(b*c).\,</math>
| |
| | |
| But if we write ''ƒ''(''a'', ''b'') instead of ''a'' * ''b'' then the associative law looks more like what one conventionally thinks of as a functional equation:
| |
| | |
| ::<math>f(f(a, b),c) = f(a, f(b, c)).\,\!</math>
| |
| | |
| One thing that all of the examples listed above share 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 substituted into the unknown function to be solved for.
| |
| | |
| *The b-integer and b-decimal parts of real numbers were introduced and studied by M.H.Hooshmand.<ref name="b-digital">{{cite journal
| |
| |author=Hooshmand, M.H.
| |
| |year=2005
| |
| |title=b-Digital sequences
| |
| |journal=[[Wmsci 2005: 9Th World Multi-Conference on Systemics, Cybernetics and Informatics]]
| |
| |volume=8
| |
| |pages=142–146
| |
| |url=http://apps.isiknowledge.com/full_record.do?product=UA&search_mode=GeneralSearch&qid=1&SID=N1gDCAgALmD8dPO9IkD&page=1&doc=4&colname=ISIP
| |
| }}</ref> The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation:
| |
| | |
| ::<math>f(f(x) + y - f(y)) = f(x).\,\!</math>
| |
| | |
| The following functional equations are as a generalization of the b-parts functional equation for semigroups and groups, even in a binary system (magma), that are introduced by him:
| |
| | |
| '''Associative equations''' ;
| |
| | |
| : <math>f(f(xy)z)=f(xf(yz))\; ,\; f(f(xy)z)=f(xf(yz))=f(xyz) </math>
| |
| | |
| '''Decomposer equations''' ;
| |
| | |
| : <math>f(f^*(x)f(y))=f(y)\; ,\; f(f(x)f_*(y))=f(x) </math>
| |
| | |
| '''Strong decomposer equations''' ;
| |
| | |
| : <math>f(f^*(x)y)=f(y)\; ,\; f(xf_*(y))=f(x) </math>
| |
| | |
| '''Canceler equations''' ;
| |
| | |
| : <math>f(f(x)y)=f(xy)\; ,\; f(xf(y))=f(xy)\; ,\; f(xf(y)z)=f(xyz) </math>
| |
| | |
| where ''ƒ''<sup> *</sup>(''x'')''ƒ''(''x'') = ''ƒ''(''x'')''ƒ''<sup> *</sup>(''x'') = ''x''.
| |
| In ref <ref name="associative">{{cite journal
| |
| |author=Hooshmand, M.H., Haili, H.K.
| |
| |year=2007
| |
| |title=Decomposer and associative functional equations
| |
| |journal=[[Indagationes Mathematicae]]
| |
| |volume=18
| |
| |issue=4
| |
| |pages=539–554
| |
| |doi= 10.1016/S0019-3577(07)80061-9
| |
| |url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6VJN-4T40TVP-5&_user=4187955&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_version=1&_urlVersion=0&_userid=4187955&md5=e4dd79d2ea4e66d4cb8159cb0501625f
| |
| |last2=Haili
| |
| |first2=H
| |
| }}</ref>
| |
| the general solution of the decomposer and strong decomposer equations are introduced in the sets with a binary operation and semigroups respectively and also associative equations in arbitrary groups. But it is proven there that the associative equations and the system of strong decomposer and canceler equations do not have any nontrivial solutions in the simple groups.
| |
| | |
| 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.
| |
| | |
| ==Solving functional equations==
| |
| Solving functional equations can be very difficult but there are some common methods of solving them. For example, 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. The main method of solving elementary functional equations is substitution. It is often useful to prove surjectivity or injectivity and prove oddness or evenness, if possible. It is also useful to guess possible solutions. Induction is a useful technique to use when the function is only defined for rational or integer values.
| |
| | |
| A discussion of [[Involution (mathematics)|involutary]] functions is useful. For example, consider the function
| |
| | |
| :<math>f(x) = 1-x \, .</math>
| |
| | |
| Composing ''f'' with itself gives
| |
| | |
| : <math>f(f(x)) = 1-(1-x) = x \, .</math>
| |
| | |
| Many other functions also satisfy the functional equation : <math>f(f(x)) = x \,</math>, including
| |
| | |
| :<math> f(x) = \frac{1}{x}\, ,</math>
| |
| :<math> f(x) = \frac{1}{1-x} + 1 \, .</math>
| |
| | |
| Example 1: Find all functions f that satisfy
| |
| : <math>f(x+y)^2 = f(x)^2 + f(y)^2\,</math>
| |
| for all <math>x,y \in \mathbb{R},</math> assuming ''ƒ'' is a [[real-valued function]].
| |
| | |
| Let ''x'' = ''y'' = 0
| |
| | |
| : <math>f(0)^2=f(0)^2+f(0)^2.\,</math>
| |
| | |
| So ''ƒ''(0)<sup>2</sup> = 0 and ''ƒ''(0) = 0.
| |
| | |
| Now, let ''y'' = −''x'':
| |
| | |
| :<math>f(x-x)^2=f(x)^2+f(-x)^2\,</math>
| |
| :<math>f(0)^2=f(x)^2+f(-x)^2\,</math>
| |
| :<math>0=f(x)^2+f(-x)^2\,</math>
| |
| | |
| A square of a real number is nonnegative, and a sum of nonnegative numbers is zero [[iff]] both numbers are 0. So ''ƒ''(x)<sup>2</sup> = 0 for all ''x'' and ''ƒ''(''x'') = 0 is the only solution.
| |
| | |
| ==See also==
| |
| *[[Functional equation (L-function)]]
| |
| *[[Bellman equation]]
| |
| *[[Dynamic programming]]
| |
| *[[Implicit function]]
| |
| | |
| ==Notes==
| |
| <references/> | |
| | |
| ==References==
| |
| *[[János Aczél (mathematician)|János Aczél]], ''Functional Equations and Their Applications'', [[Academic Press]], 1966.
| |
| *János Aczél & J. Dhombres, ''Functional Equations in Several Variables'', [[Cambridge University Press]], 1989.
| |
| *Pl. Kannappan, ''Functional Equations and Inequalities with Applications'', Springer, 2009.
| |
| *[[Marek Kuczma]], ''Introduction to the Theory of Functional Equations and Inequalities'', second edition, Birkhäuser, 2009.
| |
| *Henrik Stetkær, ''Functional Equations on Groups'', first edition, World Scientific Publishing, 2013.
| |
| | |
| == 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.
| |
| * [http://web.archive.org/web/20120227145129/http://www.imomath.com/tekstkut/funeqn_mr.pdf IMO Compendium text (archived)] on functional equations in problem solving.
| |
| | |
| {{DEFAULTSORT:Functional Equation}}
| |
| [[Category:Mathematical analysis]]
| |
| [[Category:Functional equations]]
| |