Functional equation

In mathematics, a functional equation

by the logarithmic functional equation ${\displaystyle \log(xy)=\log(x)+\log(y).}$

If the

is a function that satisfies the functional equation ${\displaystyle f(x+1)=xf(x)}$ and the initial value ${\displaystyle f(1)=1.}$ There are many functions that satisfy these conditions, but the gamma function is the unique one that is

Examples

• Fibonacci numbers
  , ${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$, where ${\displaystyle F_{0}=0}$ and ${\displaystyle F_{1}=1}$

• ${\displaystyle f(x+P)=f(x)}$, which characterizes the periodic functions

• ${\displaystyle f(x)=f(-x)}$, which characterizes the
  even functions
  , and likewise ${\displaystyle f(x)=-f(-x)}$, which characterizes the
  odd functions

• ${\displaystyle f(f(x))=g(x)}$, which characterizes the functional square roots of the function g

• ${\displaystyle f(x+y)=f(x)+f(y)\,\!}$ (
  Hamel basis
  for the real numbers
• ${\displaystyle f(x+y)=f(x)f(y),\,\!}$ satisfied by all exponential functions. Like Cauchy's additive functional equation, this too may have pathological, discontinuous solutions
• ${\displaystyle f(xy)=f(x)+f(y)\,\!}$, satisfied by all logarithmic functions and, over coprime integer arguments, additive functions
• ${\displaystyle f(xy)=f(x)f(y)\,\!}$, satisfied by all
  power functions and, over coprime integer arguments, multiplicative functions
• ${\displaystyle f(x+y)+f(x-y)=2[f(x)+f(y)]\,\!}$ (quadratic equation or parallelogram law)
• ${\displaystyle f((x+y)/2)=(f(x)+f(y))/2\,\!}$ (Jensen's functional equation)
• ${\displaystyle g(x+y)+g(x-y)=2[g(x)g(y)]\,\!}$ (d'Alembert's functional equation)
• ${\displaystyle f(h(x))=h(x+1)\,\!}$ (Abel equation)
• ${\displaystyle f(h(x))=cf(x)\,\!}$ (Schröder's equation).
• ${\displaystyle f(h(x))=(f(x))^{c}\,\!}$ (Böttcher's equation).
• ${\displaystyle f(h(x))=h'(x)f(x)\,\!}$ (Julia's equation).
• ${\displaystyle f(xy)=\sum g_{l}(x)h_{l}(y)\,\!}$ (Levi-Civita),
• ${\displaystyle f(x+y)=f(x)g(y)+f(y)g(x)\,\!}$ (sine addition formula and hyperbolic sine addition formula),
• ${\displaystyle g(x+y)=g(x)g(y)-f(y)f(x)\,\!}$ (cosine addition formula),
• ${\displaystyle g(x+y)=g(x)g(y)+f(y)f(x)\,\!}$ (hyperbolic cosine addition formula).
• The
  associative laws are functional equations. In its familiar form, the associative law is expressed by writing the binary operation in infix notation
  ,
  $${\displaystyle (a\circ b)\circ c=a\circ (b\circ c)~,}$$

  
  but if we write f(a, b) instead of a ○ b then the associative law looks more like a conventional functional equation,
  $${\displaystyle f(f(a,b),c)=f(a,f(b,c)).\,\!}$$

  

• The functional equation
  $${\displaystyle f(s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)f(1-s)}$$

  
  is satisfied by the Riemann zeta function, as proved here. The capital Γ denotes the gamma function.

• The gamma function is the unique solution of the following system of three equations:^{[citation needed]}
  • ${\displaystyle f(x)={f(x+1) \over x}}$
  • ${\displaystyle f(y)f\left(y+{\frac {1}{2}}\right)={\frac {\sqrt {\pi }}{2^{2y-1}}}f(2y)}$
  • ${\displaystyle f(z)f(1-z)={\pi \over \sin(\pi z)}}$          (Euler's reflection formula)
• The functional equation
  $${\displaystyle f\left({az+b \over cz+d}\right)=(cz+d)^{k}f(z)}$$

  
  where a, b, c, d are integers satisfying ${\displaystyle ad-bc=1}$, i.e. ${\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}}$ = 1, defines f to be a modular form of order k.

One feature that all of the examples listed above^{[clarification needed]} 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 inside the argument of the unknown functions to be solved for.^{[citation needed]}

When it comes to asking for all solutions, it may be the case that conditions from

The involutions are characterized by the functional equation ${\displaystyle f(f(x))=x}$. These appear in Babbage's functional equation (1820),^[3]

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

Other involutions, and solutions of the equation, include

• ${\displaystyle f(x)=a-x\,,}$
• ${\displaystyle f(x)={\frac {a}{x}}\,,}$ and
• ${\displaystyle f(x)={\frac {b-x}{1+cx}}~,}$

which includes the previous three as special cases or limits.

Solution

One method of solving elementary functional equations is substitution.^{[citation needed]}

Some solutions to functional equations have exploited

Some functional equations have been solved with the use of ansatzes, mathematical induction.^{[citation needed]}

Some classes of functional equations can be solved by computer-assisted techniques.^[vague][4]

In

See also

• Functional equation (L-function)
• Bellman equation
• Dynamic programming
• Implicit function
• Functional differential equation

Notes