Functional equation

In mathematics, a functional equation is an implicit way to specify some mathematical function . Often, the functional equation relatte values of a function at different arguments. Usually, the solution of a functional equation is supposed to be a holomorphic function, although some functional equations were initially established for functions of a discrete variable; see for example the Ackermann functions.

Examples

• Any periodic function has a functional equation of the form $f(x+p)=f(x)$ where p is the period.
  • Examples of periodic functions include trigonometric functions and elliptic functions.
• The gamma function has a functional equation relating $\Gamma (z)$ to $\Gamma (z-1)$.
• The Riemann zeta function has a function equation relating the value of $\zeta (s)$ to $\zeta (1-s)$.
• Superfunction $F$ of a given function $H$ is solution of functional equation $H(F(z))=F(z+1)$.
• Abel equation
• Schroeder equation

See also

• Linear equation
• Algebraic equation
• Differential equation
• Functional analysis