zeros and poles of rational function

A rational functionMathworldPlanetmath of a complex variable z may be presented by the equation

R(z)=a0zm+a1zm-1++amb0zn+b1zn-1++bn, (1)

where the numerator and the denominator are mutually irreducible polynomialsMathworldPlanetmath with complex coefficientsMathworldPlanetmath aj and bk (a0b00).  If  z=x+iy (x,y), then the real and imaginary partsDlmfMathworld of R(z) are rational functions of x and y.

When we factorize the numerator and the denominator in the ring  [z], we can write

R(z)=a0(z-α1)μ1(z-α2)μ2(z-αr)μrb0(z-β1)ν1(z-β2)ν2(z-βs)νs, (2)

where  αjβk  for all j,k.

The form (2) of the rational function expresses the zeros αj and the infinity places βk of the functionMathworldPlanetmath.  One can write (2) as


where Sj(z) is a rational function which in  z=αj  gets a finite non-zero value.  Accordingly one says that the point αj is a zero of R(z) with the order μj (j=1, 2,,r).  One can also write (2) as


where Tk(z) is a rational function getting in the point βk a finite non-zero value.  As  zβk,  the modulus |R(z)| increases unboundedly in such a manner that  |z-βk|νk|R(z)|  tends to a finite non-zero limit.  So one says that R(z) has in the point βk a pole with the order νk (k= 1, 2,,s).

Behaviour at infinity

Now let |z| increase unboundedly.  When we write


we get three cases:

  • If  m>n,  then  limzR(z)=.  Since  limzR(z)zm-n=a0b0  is finite and non-zero, the point  z=  is the pole of R(z) with the order m-n.

  • If  m=n,  we have  limzR(z)=a0b0  and thus R(z) has in the infinity a finite non-zero value.

  • If  m<n,  we have  limzR(z)=0  in such a manner that  limzzn-mR(z)=a0b0.  This means that R(z) has in infinity a zero with the order n-m.

In any case, R(z) has equally many zeros and poles, provided that each zero and pole is counted so many times as its order says.  The common number of the zeros and poles is called the order of the rational function.  It is the greatest of the degrees ( m and n of the numerator and denominator.


Denote by c any non-zero complex numberMathworldPlanetmathPlanetmath.  The c-place of R(z) means such a point z for which  R(z)=c.  If z0 is a c-place of


where the polynomialsMathworldPlanetmathPlanetmath P(z) and Q(z) have no common factor (, then z0 is a zero of

R(z)-c=P(z)-cQ(z)Q(z). (3)

If this zero is of order μ, then one says that z0 is of order μ as the c-place of R(z).  The numerator and denominator of (3) cannot have common factor (otherwise any common factor would be also a factor of P(z)).  This implies that the order of the rational function defined by (3) is the same as the order k of R(z).  Because (3) gets k times the value 0, also R(z) gets k times the value c.  Thus we have derived the

Theorem.  A rational function attains any complex value so many times as its order is.


  • 1 R. Nevanlinna & V. Paatero: Funktioteoria.  Kustannusosakeyhtiö Otava. Helsinki (1963).

Title zeros and poles of rational function
Canonical name ZerosAndPolesOfRationalFunction
Date of creation 2014-02-23 18:12:33
Last modified on 2014-02-23 18:12:33
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Topic
Classification msc 30D10
Classification msc 30C15
Classification msc 30A99
Classification msc 26C15
Related topic MinimalAndMaximalNumber
Related topic OrderValuation
Related topic RolfNevanlinna
Related topic PlacesOfHolomorphicFunction
Related topic ZeroOfPolynomial
Defines order of rational function
Defines order
Defines c-place
Defines place