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Functional Equation

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An equation of the form f(x,y,...)=0, where f contains a finite number of independent variables, known functions, and unknown functions which are to be solved for. Many properties of functions can be determined by studying the types of functional equations they satisfy. For example, the gamma function Gamma(z) satisfies the functional equations

Gamma(1+z)=zGamma(z)
(1)
Gamma(1-z)=-zGamma(-z).
(2)

The following functional equations hold

f(x)=f(x+1)+f(x^2+x+1)
(3)
l(x)=l(2x+1)+l(2x)
(4)
tau(x)=tau(x+1)+tau(x^2+x-1)
(5)
sigma(x)=sigma(sqrt(x^2+1))+sigma(xsqrt(x^2+1)(sqrt(x^2+1)+sqrt(x^2+2)))
(6)
rho(x)=rho(sqrt(x^2+1))+rho(xsqrt(x^2+1)(x+sqrt(x^2-1)))
(7)
rho(x)=rho((x^2)/((x-1)sqrt(x^2-1)+sqrt(2x+1)))-rho(x/(x-1)),
(8)

where

f(x)=tan^(-1)(1/x)
(9)
l(x)=ln(1+1/x)
(10)
tau(x)=tanh^(-1)(1/x)
(11)
=1/2ln((x-1)/(x+1))
(12)
sigma(x)=sinh^(-1)(1/x)
(13)
rho(x)=sin^(-1)(1/x)
(14)

(Borwein et al. 2004).

See also

Abel's Duplication Formula, Abel's Functional Equation, Functional Analysis, Reflection Relation

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References

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Castillo, E.; Gutiérrez, J. M.; and Iglesias, A. "Solving a Functional Equation." Mathematica J. 5, 82-86, 1995.Castillo, E. and Iglesias, A. "A Package for Symbolic Solution of Functional Equations." Mathematics with Vision: Proceedings of the First International Mathematica Symposium. pp. 85-92, 1995.Flajolet, P. and Sedgewick, R. "Analytic Combinatorics: Functional Equations, Rational and Algebraic Functions." http://www.inria.fr/RRRT/RR-4103.html.Kuczma, M. Functional Equations in a Single Variable. Warsaw, Poland: Polska Akademia Nauk, 1968.Kuczma, M. An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality. Warsaw, Poland: Uniwersitet Slaski, 1985.Kuczma, M.; Choczewski, B.; and Ger, R. Iterative Functional Equations. Cambridge, England: Cambridge University Press, 1990.

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Functional Equation

Cite this as:

Weisstein, Eric W. "Functional Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FunctionalEquation.html

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