•
Recurrence relations 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, F_{n} = F_{n-1}+F_{n-2}, where F_0=0 and F_1=1 • f(x+P) = f(x), which characterizes the
periodic functions • f(x) = f(-x), which characterizes the
even functions, and likewise f(x) = -f(-x), which characterizes the
odd functions • f(f(x)) = g(x), which characterizes the
functional square roots of a function g • f(x + y) = f(x) + f(y) (
Cauchy's functional equation), satisfied by
linear maps. 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 • f(x + y) = f(x)f(y), satisfied by all
exponential functions. Like Cauchy's additive functional equation, this too may have pathological, discontinuous solutions • f(xy) = f(x) + f(y), satisfied by all
logarithmic functions and, over coprime integer arguments,
additive functions • f(xy) = f(x) f(y), satisfied by all
power functions and, over coprime integer arguments,
multiplicative functions • f(x + y) + f(x - y) = 2[f(x) + f(y)] (quadratic equation or
parallelogram law) • f((x + y)/2) = (f(x) + f(y))/2 (
Jensen's functional equation) • g(x + y) + g(x - y) = 2[g(x) g(y)] (
d'Alembert's functional equation) • f(h(x)) = h(x + 1) (
Abel equation) • f(h(x)) = cf(x) (
Schröder's equation). • f(h(x)) = (f(x))^c (
Böttcher's equation). • f(h(x)) = h'(x)f(x) (
Julia's equation). • f(xy) = \sum g_l(x) h_l(y) (Levi-Civita), • f(x+y) = f(x)g(y)+f(y)g(x) (
sine addition formula and
hyperbolic sine addition formula), • g(x+y) = g(x)g(y)-f(y)f(x) (
cosine addition formula), • g(x+y) = g(x)g(y)+f(y)f(x) (
hyperbolic cosine addition formula). • The
commutative and
associative laws are functional equations. In its familiar form, the associative law is expressed by writing the
binary operation in
infix notation, (a \circ b) \circ c = a \circ (b \circ c), but if we write instead of then the associative law looks more like a conventional functional equation, f(f(a, b),c) = f(a, f(b, c)). • The functional equation f(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)f(1-s) is satisfied by the
Riemann zeta function. The capital denotes the
gamma function. • The gamma function is the unique solution of the following system of three equations: • f(x)={f(x+1) \over x} • f(y)f\left(y+\frac{1}{2}\right)=\frac{\sqrt{\pi}}{2^{2y-1}}f(2y) • f(z)f(1-z)={\pi \over \sin(\pi z)}(
Euler's reflection formula) • The functional equation f\left({az+b\over cz+d}\right) = (cz+d)^k f(z) where are
integers satisfying ad - bc = 1, i.e. \begin{vmatrix} a & b\\ c & d \end{vmatrix} = 1, defines to be a
modular form of order . 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 functions 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 numbers as
vector space over the
rational numbers). The
Bohr–Mollerup theorem is another well-known example.
Involutions The
involutions are characterized by the functional equation f(f(x)) = x. These appear in
Babbage's functional equation (1820), : f(f(x)) = 1-(1-x) = x \, . Other involutions, and solutions of the equation, include • f(x) = a-x\, , • f(x) = \frac{a}{x}\, , and • f(x) = \frac{b-x}{1+cx} ~ , which includes the previous three as
special cases or limits. == Solution ==