Fractals/Mathematics/Function

=Function= Properities of the function
 * number of variables ( Functions of one variable, Functions of two variables, Multivariate function )
 * type of the variable so type of input and output ( Real function, Vector-valued function, compex functions )



Notation
Types of the notation
 * functional: $$f(x)=x^2$$.
 * arrow: $$x\mapsto x^2$$



In expanded arrow notation:
 * first line defines the rule of a function inline, without requiring a name to be given to the function
 * in second line the domain and codomain is explicitly stated


 * $$\begin{align}

\operatorname{sqr}\colon \Z &\to \Z\\ x &\mapsto x^2.\end{align}$$

input
Input type:
 * number
 * position on the plane (then function is a map, iterated  function )
 * angle doubling map
 * rotating map
 * Complex quadratic map
 * rational map
 * position in the gradient ( color transfer function)
 * whole plane (then the function is the geometric plane transformation )

Specifying a function

 * By listing function values. For example, if $$A = \{ 1, 2, 3 \}$$, then one can define a function $$f\colon A \to \mathbb{R}$$ by $$f(1) = 2, f(2) = 3, f(3) = 4.$$
 * by formula
 * By recurrence relation. Functions whose domain are the nonnegative integers, known as sequences, are often defined by recurrence relations. The factorial function on the nonnegative integers ($$n\mapsto n!$$) is a basic example
 * Using differential calculus. Many functions can be defined as the antiderivative of another function. This is the case of the natural logarithm, which is the antiderivative of $1/x$ that is 0 for $x = 1$. Another common example is the error function. More generally, many functions, including most special functions, can be defined as solutions of differential equations. The simplest example is probably the exponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 for $x = 0$.

transformations

 * iteration
 * composition
 * inversion

=Types=
 * polynomial
 * rational

Rational function
Rational map f is the ratio of 2 polynomials

$$ f(z)=\frac{p(z)}{q(z)}$$

where:
 * p, q are co-prime polynomials = p and q are polynomial functions with no common zeros (if they did have a common factor, we could just cancel them) = the rational function is in reduced form
 * f is not a constant function
 * p is a numerator
 * q is the denominator and q isn't zero
 * f is a differentiable mapping from the the two-dimensional sphere (Riemann sphere)  into itself.$$f: S^2\to S^2 $$

zeros and poles

 * the zeros of rational function f(z) = the zeros of numerator p(z)
 * the poles of of rational function f(z) = the zeros of denominator q(z) = Vertical Asymptotes = the values where the denominator is equal to zero = the points where the rational function is not defined
 * the multiplicity of a zero (or pole) of rational function f is the multiplicity of the root of numerator ( or denominator ) of rational function f
 * complex poles or zeros come in complex conjugate pairs

difference between polynomial and rational function

 * infinity is always is superattracting fixed point for polynomials, for rational functions it is not true.
 * the term rational function sometimes includes polynomials ( as a degenerative or non-proper case ) and sometimes it only includes only the proper rational functions ( without polynomials

=See also=
 * derivative of the function
 * analysis of iterated map
 * Period
 * NIST Digital Library of Mathematical Functions

=References=