Fractals/Mathematics/Function

Rational functionEdit

Rational map f is the ratio of 2 polynomials[1]

  

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)[2] = 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. 


zeros and polesEdit

  • 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) [3] = Vertical Asymptotes [4] = the values where the denominator is equal to zero = the points where the rational function is not defined[5]
  • 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[6]
  • complex poles or zeros come in complex conjugate pairs

See alsoEdit

ReferencesEdit

  1. Dynamics of rational maps by Guizhen Cui, July 16, 2013
  2. Rational Functions by Kevin Wortman
  3. Izidor Hafner "3D Plots of Rational Functions of a Complex Variable" http://demonstrations.wolfram.com/3DPlotsOfRationalFunctionsOfAComplexVariable/ Wolfram Demonstrations Project Published: March 22 2016
  4. Finding the Roots & Vertical Asymptotes of Rational Functions by Cole's World of Mathematics
  5. math.stackexchange question: how-to-find-the-domain-of-a-complex-rational-function
  6. S. Boyd : EE102 Lecture 5: Rational functions and partial fraction expansion