Pole Calculator
Find poles (singularities) and their orders for rational functions.
Rational Function f(x) = P(x)/Q(x)
Poles are zeros of the denominator Q(x)
Search Parameters
Pole Types
Simple pole (order 1): f(z) ~ c/(z-z0)
Pole of order n: f(z) ~ c/(z-z0)^n
Essential singularity: Infinite order pole
Number of Poles Found
2
in range [-5, 5]
Poles Found
z = -1.000000Simple Pole
Residue: -0.500000
z = 1.000000Simple Pole
Residue: 0.500000
#Total Poles
2
sumSum of Residues
0.000000
Applications
- Partial fraction decomposition
- Stability analysis in control theory
- Residue theorem calculations
- Transfer function analysis
About Poles
Definition
A pole of a function is a point where the function approaches infinity. For rational functions, poles occur at zeros of the denominator that are not canceled by the numerator.
Order of Poles
The order of a pole is the multiplicity of the zero in the denominator. A simple pole has order 1, while higher-order poles have the form 1/(z-z0)^n.