Functional equation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a functional equation
[irrelevant citation] is, in the broadest meaning, an <a href="/facts/Equation/pxFzLAVR">equation</a> in which one or several functions appear as <a href="/facts/Unknown_(mathematics)/pxFzLAVR">unknowns</a>. So, <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> and <a href="/facts/Integral_equation/LCu4Imfq">integral equations</a> 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 <a href="/facts/Logarithm_function/QeXaV97q">logarithm functions</a> are <a href="/facts/Logarithm/QeXaV97q">essentially characterized</a> by the logarithmic functional equation 
 
 
 
 log
 ⁡
 (
 x
 y
 )
 =
 log
 ⁡
 (
 x
 )
 +
 log
 ⁡
 (
 y
 )
 .
 
 
 {\displaystyle \log(xy)=\log(x)+\log(y).}

If the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> of the unknown function is supposed to be the <a href="/facts/Natural_number/u65og8JE">natural numbers</a>, the function is generally viewed as a <a href="/facts/Sequence_(mathematics)/gV2S4uqp">sequence</a>, and, in this case, a functional equation (in the narrower meaning) is called a <a href="/facts/Recurrence_relation/JolXC71M">recurrence relation</a>. Thus the term functional equation is used mainly for <a href="/facts/Real_function/bbNG3DnN">real functions</a> and <a href="/facts/Complex_function/hgCoQlH0">complex functions</a>. Moreover a <a href="/facts/Smooth_function/mffL4ch2">smoothness condition</a> is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> is a function that satisfies the functional equation 
 
 
 
 f
 (
 x
 +
 1
 )
 =
 x
 f
 (
 x
 )
 
 
 {\displaystyle f(x+1)=xf(x)}
 
 and the initial value 
 
 
 
 f
 (
 1
 )
 =
 1.
 
 
 {\displaystyle f(1)=1.}
 
 There are many functions that satisfy these conditions, but the gamma function is the unique one that is <a href="/facts/Meromorphic_function/LeNBd6Sh">meromorphic</a> in the whole complex plane, and <a href="/facts/Logarithmically_convex_function/1fyzx7Ig">logarithmically convex</a> for x real and positive (<a href="/facts/Bohr%25E2%2580%2593Mollerup_theorem/p5qpCz7m">Bohr–Mollerup theorem</a>).

Functional equation open-in-new

Functional equation