# Rational function

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

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


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)[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.${\displaystyle f:S^{2}\to S^{2}}$

## 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) [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