Functional equation
As a functional equation, an equation is called in mathematics, one or more functions are sought for their solution. Many functions can be defined on an underlying functional equation. Usually referred to as functional equations, only those equations which can not be brought about by transformations on an explicit closed form for the unknown function (s), and where the unknown function with different arguments occurs.
In the investigation of functional equations one is interested in all solution functions of the examined functional space, not only on one. Otherwise, it's pretty trivial to any given function to construct a functional equation.
"It is natural to ask what a functional equation is. But there is no easy answer to this question satisfactory. "
- 3.1 recurrences
- 3.2 computational rules
From Cauchy examined functional equations
Augustin Louis Cauchy in 1821 examined the steady solutions of the following functional equations in his Cours d'analyze de l' Ecole Royale Polytechnique, Chapter 5:
The steady solutions of this functional equation are the linear functions, with a real constant. For this functional equation, the term Cauchy ( cal ) functional equation has naturalized.
The steady solutions of this functional equation are the power functions, with a real constant.
The steady solutions of this functional equation are the exponential functions, with a positive real constant.
The steady solutions of this functional equation are the logarithmic functions, with a positive real constant.
Furthermore, the zero function is a trivial solution of each of these functional equations.
Known functional equations of special functions
Gamma function
The functional equation
Is met by the gamma function. If we consider only functions which are log convex, then all solutions of this equation described by, with. This is the set of Bohr - Mollerup about the uniqueness of the gamma function as a continuation of the faculties from to.
Furthermore, the gamma function is also a solution of the functional equation
Only a special kind of " reflection symmetry " to represent how by means of the substitution and then the logarithm of the new functional equation looks.
Polygammafunktionen
For the functional equations are
Satisfied by the Polygammafunktionen. For fixed all continuous and monotone solutions are represented by the functions with arbitrary.
Bernoulli polynomials
For the functional equations are
Satisfied by the Bernoulli polynomials. All steady solutions of this equation are described by plus another (periodic ) solutions of the homogeneous functional equation, where a is an arbitrary real number. More details about this in the following section.
Periodic functions
The functional equation
Represents the homogeneous Lösungensanteil the above functional equations, whose solution simply because you can add to a solution of some inhomogeneous functional equation and thus receives a new, as long as you hurt no more restrictive conditions. Considering all holomorphic functions on all, so all solution functions are represented by
This knowledge is the basis of Fourier analysis. All these functions are except for the case n = 0 neither convex nor monotone.
Zeta function
The functional equation
Is satisfied by the Riemann zeta function. denotes the gamma function.
Note: Due to the substitution
And subsequent algebraic simplification, this functional equation is for transferred into a new one for that
Reads. Thus, the original functional equation can be brought to a shape by the transformation, which only requires a function of reading. The transformed accordingly so Riemann zeta function is known as the Riemann Xi - function.
Even and odd functions
The two functional equations
Be met by all even and odd functions. Another "simple" function equation is
So all functions that are their own inverse function on the interval, describe their solution set. These three functional equations but is rather on the question of how their solutions can be characterized usefully.
" Real " iterate of a function
Given an analytic, bijective function, is Schroeder's functional equation
With a firm determined to. Applying to both sides of this equation by the inverse function, then one can generalize the definition of
And for any fixed t, this function behaves like a t -fold iterated function. A simple example is given for fixed general power function for on. In this case, the solution of Schröder 's equation and with the result, that is.
Modular forms
The functional equation
Being added, is used in the definition of module types.
Wavelets and Approximation Theory
For and defines the functional equation
In the theory of Waveletbasen the scaling function of a multiresolution analysis. The important in approximation theory and computer graphics B -splines are solutions of such a refinement equation, other solutions, including the coefficients can be found at Daubechies wavelets. There are extensions with vektorwertigem solution functions f and matrices as coefficients.
Sine and cosine
Considering the functional equation which satisfies the exponential function over the complex numbers and divides the range of values into real and imaginary parts, ie, and also restricts the domain of a, we obtain two functional equations in two unknown functions, namely
And
This corresponds to the addition theorems and can be considered as a functional equation for the real system, the sine and cosine functions.
Other examples of general functional equations
Recurrence equations
A simple class of functional equations is over from the recurrence equations. Formally there an unknown function is sought.
A very simple example of such recurrence is about the Fibonacci sequence, the linear equation:
This can of course also consider embedded in the set of real numbers, so here
Whose analytical solutions then all the form
Have with any. Just as a function can be all of their functions, for example as a solution
Specify. Abwohl irrational numbers occur in this representation is obtained for each an integer value as long as is.
Computational rules
Arithmetic laws as commutative, associative law and distributive law can also be interpreted as functional equations.
Example associative law: Given an associative amount. For their binary logic or two-parameter function apply to all
And in
Being identified.
Denote the binary logic function 2nd stage (eg multiplication), and link function 1ster level (eg addition), then a distributive law would be written as a functional equation
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Comments
All examples have in common that two or more known functions ( multiplication by a constant, addition, or just the identical function ) are used as arguments of the unknown function.
When searching for all solutions of a functional equation often additional conditions are imposed, for example, continuity is in the aforementioned Cauchy equation required for reasonable solutions. Georg Hamel, however, has shown in 1905 that, given the axiom of choice, there are also discontinuous solutions. These solutions are based on a Hamel basis of the real numbers as a vector space over the rational numbers and are mainly of theoretical importance.