FunctionPoles

FunctionPoles[f,x]

finds the poles of the meromorphic function f with the variable x.

FunctionPoles[{f,cons},x]

gives the poles of f when x is restricted by the constraints cons.

Details

  • Function poles are also known as pole singularities.
  • Function poles are often used to compute the residue of a function in complex analysis or to compute the radius of convergence for a power series.
  • A function has a pole singularity at with multiplicity if it has a series representation of the form . A function is meromorphic if it only has pole singularities.
  • FunctionPoles returns a list of pairs {pole,multiplicity}.
  • 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$Assumptionsassumptions on parameters
    GeneratedParametersChow to name parameters that are generated
    PerformanceGoal$PerformanceGoalwhether to prioritize speed or quality
  • Some of the returned poles may have Indeterminate multiplicity if FunctionPoles fails to determine their multiplicity.

Examples

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Basic Examples  (2)

Find poles of a rational function:

The set of poles may be infinite:

Scope  (6)

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:

Options  (3)

Assumptions  (1)

Specify conditions on parameters:

GeneratedParameters  (1)

FunctionPoles may introduce new parameters to represent the solution:

Use GeneratedParameters to control how the parameters are named:

PerformanceGoal  (1)

Computing multiplicities of poles may take a long time:

PerformanceGoal"Speed" limits the time allowed for computation of multiplicity:

The poles returned in both cases are the same:

In the first case, all multiplicities are computed successfully:

In the second case, some multiplicities are not computed:

Applications  (3)

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:

Properties & Relations  (4)

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:

Possible Issues  (2)

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:

Wolfram Research (2021), FunctionPoles, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionPoles.html.

Text

Wolfram Research (2021), FunctionPoles, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionPoles.html.

CMS

Wolfram Language. 2021. "FunctionPoles." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionPoles.html.

APA

Wolfram Language. (2021). FunctionPoles. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionPoles.html

BibTeX

@misc{reference.wolfram_2022_functionpoles, author="Wolfram Research", title="{FunctionPoles}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionPoles.html}", note=[Accessed: 03-July-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_functionpoles, organization={Wolfram Research}, title={FunctionPoles}, year={2021}, url={https://reference.wolfram.com/language/ref/FunctionPoles.html}, note=[Accessed: 03-July-2022 ]}