Functional equation

<h2 id="examples">Examples</h2>
<ul><li><a href="/facts/Recurrence_relation/JolXC71M">Recurrence relations</a> 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 <a href="/facts/Shift_operator/6c6lc0ks">shift operator</a>. For example, the recurrence relation defining the <a href="/facts/Fibonacci_numbers/mqxpAUQD">Fibonacci numbers</a>, 
 
 
 
 
 F
 
 n
 
 
 =
 
 F
 
 n
 −
 1
 
 
 +
 
 F
 
 n
 −
 2
 
 
 
 
 {\displaystyle F_{n}=F_{n-1}+F_{n-2}}
 
, where 
 
 
 
 
 F
 
 0
 
 
 =
 0
 
 
 {\displaystyle F_{0}=0}
 
 and 
 
 
 
 
 F
 
 1
 
 
 =
 1
 
 
 {\displaystyle F_{1}=1}
 
</li></ul>
<ul><li>
 
 
 
 f
 (
 x
 +
 P
 )
 =
 f
 (
 x
 )
 
 
 {\displaystyle f(x+P)=f(x)}
 
, which characterizes the <a href="/facts/Periodic_function/Sm8lJQsF">periodic functions</a></li></ul>
<ul><li>
 
 
 
 f
 (
 x
 )
 =
 f
 (
 −
 x
 )
 
 
 {\displaystyle f(x)=f(-x)}
 
, which characterizes the <a href="/facts/Even_function/tGHWJqAa">even functions</a>, and likewise 
 
 
 
 f
 (
 x
 )
 =
 −
 f
 (
 −
 x
 )
 
 
 {\displaystyle f(x)=-f(-x)}
 
, which characterizes the <a href="/facts/Odd_function/tGHWJqAa">odd functions</a></li></ul>
<ul><li>
 
 
 
 f
 (
 f
 (
 x
 )
 )
 =
 g
 (
 x
 )
 
 
 {\displaystyle f(f(x))=g(x)}
 
, which characterizes the <a href="/facts/Functional_square_root/C3qCTVCw">functional square roots</a> of the function g</li></ul>
<ul><li>
 
 
 
 f
 (
 x
 +
 y
 )
 =
 f
 (
 x
 )
 +
 f
 (
 y
 )
 
 
 
 
 {\displaystyle f(x+y)=f(x)+f(y)\,\!}
 
 (<a href="/facts/Cauchy%2527s_functional_equation/PHkY7R3F">Cauchy's functional equation</a>), satisfied by <a href="/facts/Linear_map/Q1PBKNVV">linear maps</a>. The equation may, contingent on the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a>, also have other pathological nonlinear solutions, whose existence can be proven with a <a href="/facts/Hamel_basis/89IPoN6c">Hamel basis</a> for the real numbers</li>
<li>
 
 
 
 f
 (
 x
 +
 y
 )
 =
 f
 (
 x
 )
 f
 (
 y
 )
 ,
 
 
 
 
 {\displaystyle f(x+y)=f(x)f(y),\,\!}
 
 satisfied by all <a href="/facts/Exponential_function/ewlYxvR2">exponential functions</a>. Like Cauchy's additive functional equation, this too may have pathological, discontinuous solutions</li>
<li>
 
 
 
 f
 (
 x
 y
 )
 =
 f
 (
 x
 )
 +
 f
 (
 y
 )
 
 
 
 
 {\displaystyle f(xy)=f(x)+f(y)\,\!}
 
, satisfied by all <a href="/facts/Logarithm/QeXaV97q">logarithmic</a> functions and, over coprime integer arguments, <a href="/facts/Additive_function/J5XLJSz4">additive functions</a></li>
<li>
 
 
 
 f
 (
 x
 y
 )
 =
 f
 (
 x
 )
 f
 (
 y
 )
 
 
 
 
 {\displaystyle f(xy)=f(x)f(y)\,\!}
 
, satisfied by all <a href="/facts/Power_function/pac6QY2g">power functions</a> and, over coprime integer arguments, <a href="/facts/Multiplicative_function/4kfolbBl">multiplicative functions</a></li>
<li>
 
 
 
 f
 (
 x
 +
 y
 )
 +
 f
 (
 x
 −
 y
 )
 =
 2
 [
 f
 (
 x
 )
 +
 f
 (
 y
 )
 ]
 
 
 
 
 {\displaystyle f(x+y)+f(x-y)=2[f(x)+f(y)]\,\!}
 
 (quadratic equation or <a href="/facts/Parallelogram_law/JI1LwPxd">parallelogram law</a>)</li>
<li>
 
 
 
 f
 (
 (
 x
 +
 y
 )
 
 /
 
 2
 )
 =
 (
 f
 (
 x
 )
 +
 f
 (
 y
 )
 )
 
 /
 
 2
 
 
 
 
 {\displaystyle f((x+y)/2)=(f(x)+f(y))/2\,\!}
 
 (Jensen's functional equation)</li>
<li>
 
 
 
 g
 (
 x
 +
 y
 )
 +
 g
 (
 x
 −
 y
 )
 =
 2
 [
 g
 (
 x
 )
 g
 (
 y
 )
 ]
 
 
 
 
 {\displaystyle g(x+y)+g(x-y)=2[g(x)g(y)]\,\!}
 
 (d'Alembert's functional equation)</li>
<li>
 
 
 
 f
 (
 h
 (
 x
 )
 )
 =
 h
 (
 x
 +
 1
 )
 
 
 
 
 {\displaystyle f(h(x))=h(x+1)\,\!}
 
 (<a href="/facts/Abel_equation/VR16bsDB">Abel equation</a>)</li>
<li>
 
 
 
 f
 (
 h
 (
 x
 )
 )
 =
 c
 f
 (
 x
 )
 
 
 
 
 {\displaystyle f(h(x))=cf(x)\,\!}
 
 (<a href="/facts/Schr%25C3%25B6der%2527s_equation/jWWTbrra">Schröder's equation</a>).</li>
<li>
 
 
 
 f
 (
 h
 (
 x
 )
 )
 =
 (
 f
 (
 x
 )
 
 )
 
 c
 
 
 
 
 
 
 {\displaystyle f(h(x))=(f(x))^{c}\,\!}
 
 (<a href="/facts/B%25C3%25B6ttcher%2527s_equation/XDwY9GPJ">Böttcher's equation</a>).</li>
<li>
 
 
 
 f
 (
 h
 (
 x
 )
 )
 =
 
 h
 ′
 
 (
 x
 )
 f
 (
 x
 )
 
 
 
 
 {\displaystyle f(h(x))=h'(x)f(x)\,\!}
 
 (<a href="/facts/Schr%25C3%25B6der%2527s_equation/jWWTbrra">Julia's equation</a>).</li>
<li>
 
 
 
 f
 (
 x
 y
 )
 =
 ∑
 
 g
 
 l
 
 
 (
 x
 )
 
 h
 
 l
 
 
 (
 y
 )
 
 
 
 
 {\displaystyle f(xy)=\sum g_{l}(x)h_{l}(y)\,\!}
 
 (Levi-Civita),</li>
<li>
 
 
 
 f
 (
 x
 +
 y
 )
 =
 f
 (
 x
 )
 g
 (
 y
 )
 +
 f
 (
 y
 )
 g
 (
 x
 )
 
 
 
 
 {\displaystyle f(x+y)=f(x)g(y)+f(y)g(x)\,\!}
 
 (<a href="/facts/List_of_trigonometric_identities/Xw9Jx8Vz">sine addition formula</a> and <a href="/facts/Hyperbolic_functions/Y4Cb5S35">hyperbolic sine addition formula</a>),</li>
<li>
 
 
 
 g
 (
 x
 +
 y
 )
 =
 g
 (
 x
 )
 g
 (
 y
 )
 −
 f
 (
 y
 )
 f
 (
 x
 )
 
 
 
 
 {\displaystyle g(x+y)=g(x)g(y)-f(y)f(x)\,\!}
 
 (<a href="/facts/List_of_trigonometric_identities/Xw9Jx8Vz">cosine addition formula</a>),</li>
<li>
 
 
 
 g
 (
 x
 +
 y
 )
 =
 g
 (
 x
 )
 g
 (
 y
 )
 +
 f
 (
 y
 )
 f
 (
 x
 )
 
 
 
 
 {\displaystyle g(x+y)=g(x)g(y)+f(y)f(x)\,\!}
 
 (<a href="/facts/Hyperbolic_functions/Y4Cb5S35">hyperbolic cosine addition formula</a>).</li>
<li>The <a href="/facts/Commutative_law/WaYxJLxd">commutative</a> and <a href="/facts/Associative_law/EQX8lV6r">associative laws</a> are functional equations. In its familiar form, the associative law is expressed by writing the <a href="/facts/Binary_operation/CYtow0bz">binary operation</a> in <a href="/facts/Infix_notation/LY0dMhay">infix notation</a>, 
 
 
 
 (
 a
 ∘
 b
 )
 ∘
 c
 =
 a
 ∘
 (
 b
 ∘
 c
 )
  
 ,
 
 
 {\displaystyle (a\circ b)\circ c=a\circ (b\circ c)~,}
 
 but if we write f(a, b) instead of a ○ b then the associative law looks more like a conventional functional equation, 
 
 
 
 f
 (
 f
 (
 a
 ,
 b
 )
 ,
 c
 )
 =
 f
 (
 a
 ,
 f
 (
 b
 ,
 c
 )
 )
 .
 
 
 
 
 {\displaystyle f(f(a,b),c)=f(a,f(b,c)).\,\!}
 
</li></ul>
<ul><li>The functional equation 
 
 
 
 f
 (
 s
 )
 =
 
 2
 
 s
 
 
 
 π
 
 s
 −
 1
 
 
 sin
 ⁡
 
 (
 
 
 
 π
 s
 
 2
 
 
 )
 
 Γ
 (
 1
 −
 s
 )
 f
 (
 1
 −
 s
 )
 
 
 {\displaystyle f(s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)f(1-s)}
 
 is satisfied by the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>, as proved <a href="/facts/Riemann_zeta_function/MnfRu7lY">here</a>. The capital Γ denotes the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>.</li></ul>
<ul><li>The gamma function is the unique solution of the following system of three equations:
<ul><li>
 
 
 
 f
 (
 x
 )
 =
 
 
 
 f
 (
 x
 +
 1
 )
 
 x
 
 
 
 
 {\displaystyle f(x)={f(x+1) \over x}}
 
</li>
<li>
 
 
 
 f
 (
 y
 )
 f
 
 (
 
 y
 +
 
 
 1
 2
 
 
 
 )
 
 =
 
 
 
 π
 
 
 2
 
 2
 y
 −
 1
 
 
 
 
 f
 (
 2
 y
 )
 
 
 {\displaystyle f(y)f\left(y+{\frac {1}{2}}\right)={\frac {\sqrt {\pi }}{2^{2y-1}}}f(2y)}
 
</li>
<li>
 
 
 
 f
 (
 z
 )
 f
 (
 1
 −
 z
 )
 =
 
 
 π
 
 sin
 ⁡
 (
 π
 z
 )
 
 
 
 
 
 {\displaystyle f(z)f(1-z)={\pi \over \sin(\pi z)}}
 
          (<a href="/facts/Leonhard_Euler/z2fsbjgj">Euler's</a> <a href="/facts/Reflection_formula/aKUhTbH0">reflection formula</a>)</li></ul></li>
<li>The functional equation 
 
 
 
 f
 
 (
 
 
 
 a
 z
 +
 b
 
 
 c
 z
 +
 d
 
 
 
 )
 
 =
 (
 c
 z
 +
 d
 
 )
 
 k
 
 
 f
 (
 z
 )
 
 
 {\displaystyle f\left({az+b \over cz+d}\right)=(cz+d)^{k}f(z)}
 
 where a, b, c, d are <a href="/facts/Integer/PUwwrawx">integers</a> satisfying 
 
 
 
 a
 d
 −
 b
 c
 =
 1
 
 
 {\displaystyle ad-bc=1}
 
, i.e. 
 
 
 
 
 
 |
 
 
 
 a
 
 
 b
 
 
 
 
 c
 
 
 d
 
 
 
 |
 
 
 
 
 {\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}}
 
 = 1, defines f to be a <a href="/facts/Modular_form/PcnbYR1f">modular form</a> of order k.</li></ul>
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 <a href="/facts/Identity_function/9AtsP0LC">identity function</a>) 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 <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a> should be applied; for example, in the case of the Cauchy equation mentioned above, the solutions that are <a href="/facts/Continuous_function/sKbl02pB">continuous functions</a> are the 'reasonable' ones, while other solutions that are not likely to have practical application can be constructed (by using a <a href="/facts/Hamel_basis/89IPoN6c">Hamel basis</a> for the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> as <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>). The <a href="/facts/Bohr%25E2%2580%2593Mollerup_theorem/p5qpCz7m">Bohr–Mollerup theorem</a> is another well-known example.

<h3>Involutions</h3>
The <a href="/facts/Involution_(mathematics)/kKxYrUVa">involutions</a> are characterized by the functional equation 
 
 
 
 f
 (
 f
 (
 x
 )
 )
 =
 x
 
 
 {\displaystyle f(f(x))=x}
 
. These appear in <a href="/facts/Charles_Babbage/T3jklQcd">Babbage's</a> functional equation (1820),<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

f
        (
        f
        (
        x
        )
        )
        =
        1
        −
        (
        1
        −
        x
        )
        =
        x
        
        .
      
    
    {\displaystyle f(f(x))=1-(1-x)=x\,.}

Other involutions, and solutions of the equation, include

<ul><li>
 
 
 
 f
 (
 x
 )
 =
 a
 −
 x
 
 ,
 
 
 {\displaystyle f(x)=a-x\,,}
 
</li>
<li>
 
 
 
 f
 (
 x
 )
 =
 
 
 a
 x
 
 
 
 ,
 
 
 {\displaystyle f(x)={\frac {a}{x}}\,,}
 
 and</li>
<li>
 
 
 
 f
 (
 x
 )
 =
 
 
 
 b
 −
 x
 
 
 1
 +
 c
 x
 
 
 
  
 ,
 
 
 {\displaystyle f(x)={\frac {b-x}{1+cx}}~,}
 
</li></ul>
which includes the previous three as <a href="/facts/Special_case/bKaJ85XJ">special cases</a> or limits.

<h2 id="solution">Solution</h2>
One method of solving elementary functional equations is substitution.
Some solutions to functional equations have exploited <a href="/facts/Surjective/KezhLpCb">surjectivity</a>, <a href="/facts/Injective_function/5ES6tqf5">injectivity</a>, <a href="/facts/Odd_function/tGHWJqAa">oddness</a>, and <a href="/facts/Even_function/tGHWJqAa">evenness</a>.
Some functional equations have been solved with the use of <a href="/facts/Ansatz/3epoB3Rg">ansatzes</a>, <a href="/facts/Mathematical_induction/KxAPqNrH">mathematical induction</a>.
Some classes of functional equations can be solved by computer-assisted techniques.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
In <a href="/facts/Dynamic_programming/CLcabtmk">dynamic programming</a> a variety of successive approximation methods<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> are used to solve <a href="/facts/Bellman_equation/Qh2gKSMH">Bellman's functional equation</a>, including methods based on <a href="/facts/Fixed_point_iteration/05dQGazd">fixed point iterations</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Functional_equation_(L-function)/qKa0wBC5">Functional equation (L-function)</a></li>
<li><a href="/facts/Bellman_equation/Qh2gKSMH">Bellman equation</a></li>
<li><a href="/facts/Dynamic_programming/CLcabtmk">Dynamic programming</a></li>
<li><a href="/facts/Implicit_function/UJmQ2Cyk">Implicit function</a></li>
<li><a href="/facts/Functional_differential_equation/sKI6hYsp">Functional differential equation</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/J%25C3%25A1nos_Acz%25C3%25A9l_(mathematician)/A5hjWDRW">János Aczél</a>, <a href="https://books.google.com/books?id=JEB0BFvRwrcC">Lectures on Functional Equations and Their Applications</a>, <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>, 1966, reprinted by Dover Publications, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0486445232.</li>
<li>János Aczél & J. Dhombres, <a href="https://books.google.com/books?id=8EWnEh18rVgC&q=%22Functional+Equations+in+Several+Variables%22">Functional Equations in Several Variables</a>, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, 1989.</li>
<li>C. Efthimiou, Introduction to Functional Equations, AMS, 2011, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-5314-6 ; <a href="https://web.archive.org/web/20230603025341/http://www.msri.org/people/staff/levy/files/MCL/Efthimiou/100914book.pdf">online</a>.</li>
<li>Pl. Kannappan, <a href="https://books.google.com/books?id=SdZoCM2OeuIC&dq=%22Functional+Equations+and+Inequalities+with+Applications%22&pg=PA1">Functional Equations and Inequalities with Applications</a>, Springer, 2009.</li>
<li><a href="/facts/Marek_Kuczma/YXmvv7R5">Marek Kuczma</a>, <a href="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</a>, second edition, Birkhäuser, 2009.</li>
<li>Henrik Stetkær, <a href="https://books.google.com/books?id=JzS7CgAAQBAJ&dq=%22Functional+Equations+on+Groups%22&pg=PR5">Functional Equations on Groups</a>, first edition, World Scientific Publishing, 2013.</li>
<li>Christopher G. Small (3 April 2007). <a href="https://books.google.com/books?id=2D2RYbb22nMC">Functional Equations and How to Solve Them</a>. Springer Science & Business Media. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-48901-8.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://eqworld.ipmnet.ru/en/solutions/fe.htm">Functional Equations: Exact Solutions</a> at EqWorld: The World of Mathematical Equations.</li>
<li><a href="http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-fe.htm">Functional Equations: Index</a> at EqWorld: The World of Mathematical Equations.</li>
<li><a href="https://web.archive.org/web/20120227145129/http://www.imomath.com/tekstkut/funeqn_mr.pdf">IMO Compendium text (archived)</a> on functional equations in problem solving.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Rassias, Themistocles M. (2000). Functional Equations and Inequalities. 3300 AA Dordrecht, The Netherlands: Kluwer Academic Publishers. p. 335. ISBN 0-7923-6484-8.{{cite book}}: CS1 maint: location (link) <a href="0-7923-6484-8" target="_blank">0-7923-6484-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Czerwik, Stephan (2002). Functional Equations and Inequalities in Several Variables. P O Box 128, Farrer Road, Singapore 912805: World Scientific Publishing Co. p. 410. ISBN 981-02-4837-7.{{cite book}}: CS1 maint: location (link) <a href="981-02-4837-7" target="_blank">981-02-4837-7</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Ritt, J. F. (1916). "On Certain Real Solutions of Babbage's Functional Equation". The Annals of Mathematics. 17 (3): 113–122. doi:10.2307/2007270. JSTOR 2007270. <a href="/wiki/Joseph_Ritt" target="_blank">/wiki/Joseph_Ritt</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Házy, Attila (2004-03-01). "Solving linear two variable functional equations with computer". Aequationes Mathematicae. 67 (1): 47–62. doi:10.1007/s00010-003-2703-9. ISSN 1420-8903. S2CID 118563768. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Bellman, R. (1957). Dynamic Programming, Princeton University Press. <a href="/wiki/Princeton_University_Press" target="_blank">/wiki/Princeton_University_Press</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Sniedovich, M. (2010). Dynamic Programming: Foundations and Principles, Taylor & Francis. <a href="/wiki/Taylor_%26_Francis" target="_blank">/wiki/Taylor_%26_Francis</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Functional equation open-in-new

Functional equation