# 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 , 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 , 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: