Functional equation

Source: Wikipedia, the free encyclopedia.

In mathematics, a functional equation

logarithm functions are essentially characterized
by the logarithmic functional equation

If the

smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the gamma function
is a function that satisfies the functional equation and the initial value There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for x real and positive (Bohr–Mollerup theorem).

Examples

  • Fibonacci numbers
    , , where and
  • , which characterizes the periodic functions
  • , which characterizes the
    even functions
    , and likewise , which characterizes the
    odd functions
  • , which characterizes the functional square roots of the function g
  • The gamma function is the unique solution of the following system of three equations:[citation needed]
    •           (Euler's reflection formula)
  • The functional equation
    where a, b, c, d are integers satisfying , i.e. = 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

is another well-known example.

Involutions

The involutions are characterized by the functional equation . These appear in Babbage's functional equation (1820),[3]

Other involutions, and solutions of the equation, include

  • and

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

evenness.[citation needed
]

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

fixed point iterations
.

See also

Notes

  1. ISBN 0-7923-6484-8.{{cite book}}: CS1 maint: location (link
    )
  2. ISBN 981-02-4837-7.{{cite book}}: CS1 maint: location (link
    )
  3. .
  4. .
  5. ^ Bellman, R. (1957). Dynamic Programming, Princeton University Press.
  6. ^ Sniedovich, M. (2010). Dynamic Programming: Foundations and Principles, Taylor & Francis.

References

External links