zeros and poles of rational function
When we factorize the numerator and the denominator in the ring , we can write
where for all .
where is a rational function which in gets a finite non-zero value. Accordingly one says that the point is a zero of with the order (). One can also write (2) as
where is a rational function getting in the point a finite non-zero value.
As , the modulus increases unboundedly in such a manner that tends to a finite non-zero limit. So one says that has in the point a pole with the order ().
Behaviour at infinity
Now let increase unboundedly. When we write
we get three cases:
If , then . Since is finite and non-zero, the point is the pole of with the order .
If , we have and thus has in the infinity a finite non-zero value.
If , we have in such a manner that . This means that has in infinity a zero with the order .
In any case, 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 (http://planetmath.org/PolynomialRing) and of the numerator and denominator.
Denote by any non-zero complex number. The -place of means such a point for which . If is a -place of
where the polynomials and have no common factor (http://planetmath.org/DivisibilityInRings), then is a zero of
If this zero is of order , then one says that is of order as the -place of . The numerator and denominator of (3) cannot have common factor (otherwise any common factor would be also a factor of ). This implies that the order of the rational function defined by (3) is the same as the order of . Because (3) gets times the value , also gets times the value . 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|
|Date of creation||2014-02-23 18:12:33|
|Last modified on||2014-02-23 18:12:33|
|Last modified by||pahio (2872)|
|Defines||order of rational function|