Find poles of a rational function:

The set of poles may be infinite:

A rational function:

A function with infinitely many poles:

Find the poles with TemplateBox[{x}, Abs]<=2:

Analytic functions have no poles:

has a removable singularity at :

FunctionPoles requires the input function to be meromorphic:

The function is meromorphic for :

Some of the returned poles may have Indeterminate multiplicity if determining the multiplicity fails:

Classify singularities of a meromorphic function:

FunctionSingularities gives locations of poles and removable singularities:

has a double pole at , a single pole at and a removable singularity at :

Integrate TemplateBox[{{{x, ^, 3}, -, {x, /, 3}}}, Gamma] along the unit circle:

Compute the poles of in the unit disk:

Compute the integral using the residue theorem:

Compare with the result of numeric integration:

Find the radius of convergence of the Taylor series of at :

The radius of convergence equals the distance to the nearest pole:

Even though the poles are complex, the convergence over the reals is affected:

Since is farther from the poles than , the convergence radius at is greater:

The limit of the absolute value of a function at a pole is :

Use Limit to compute the limit:

The first term of the power series of a function at a pole of multiplicity has exponent :

Use Series to compute the series:

Use Residue to find the coefficient at the series term with exponent :

The only singularities a meromorphic function can have are poles and removable singularities:

Use FunctionSingularities to find a condition satisfied by all singularities:

Use SolveValues to find the singularities:

The function has poles at and and a removable singularity at :

Use FunctionPoles to find the poles of a function:

Use Residue to find the residues at the poles:

ResidueSum gives the sum of the residues at all poles:

Some of the returned poles may have Indeterminate multiplicity if determining the multiplicity fails:

FunctionPoles rationalizes inexact inputs and then approximates the result to the input precision:

The result may depend on which rational numbers are chosen:

In the first example, the rationalized exponent is an integer; in the second example, it is not: