ResidueSum

ResidueSum[f,z]

finds the sum of residues of the meromorphic function f with the variable z.

ResidueSum[{f,cons},z]

finds the sum of residues of f within the solution set of the constraints cons.

Details and Options

ResidueSum computes the sum of residues at all poles of f. The residue of f at a pole z₀ is defined as the coefficient of in the Laurent expansion of f.
Sums of residues are often used to compute contour integrals using Cauchy's residue theorem.

The function f should be meromorphic for x satisfying the constraints cons.
cons can contain equations, inequalities or logical combinations of these.
The following options can be given:

Assumptions	$Assumptions	assumptions on parameters
GenerateConditions	Automatic	whether to generate conditions on parameters
PerformanceGoal	$PerformanceGoal	whether to prioritize speed or quality

Examples

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Basic Examples (2)

Find the sum of residues of a rational function:

Find the sum of residues of in a disk:

Scope (6)

Sum of residues of a rational function:

Sum of residues of a meromorphic function in a region:

Sum of residues at infinitely many poles may be finite:

Analytic functions have no poles:

has a removable singularity at :

ResidueSum requires the input function to be meromorphic:

The function is meromorphic for :

Options (4)

Assumptions (1)

Specify conditions on parameters:

GenerateConditions (2)

By default, ResidueSum may generate conditions on symbolic parameters:

With GenerateConditionsNone, ResidueSum fails instead of giving a conditional result:

This returns a conditionally valid result without stating the condition:

By default, conditions that are generically true are not reported:

With GenerateConditions->True, all conditions are reported:

PerformanceGoal (1)

Use PerformanceGoal to avoid potentially expensive computations:

The default setting uses all available techniques to try to produce a result:

Applications (1)

Integrate along the unit circle:

Compute the sum of residues of in the unit disk:

Compute the integral using the residue theorem:

Compare with the result of numeric integration:

Properties & Relations (3)

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:

Use FunctionMeromorphic to test whether a function is meromorphic:

Compute the sum of residues in a region where the function is meromorphic:

Use FunctionAnalytic to test whether a function is complex analytic:

Sum of residues of an analytic function is zero:

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ResidueSum

Details and Options

Examples

Basic Examples (2)

Scope (6)