finds the singularities of for x∈dom.
finds the singularities of for x1,x2,…∈dom.
- Function singularities are typically used to either find regions where a function is guaranteed to be analytic or to find points and curves where special analysis needs to be performed.
- FunctionSingularities gives an implicit description of a set such that is analytic in . The set is not guaranteed to be minimal.
- The resulting implicit description consists of equations, inequalities, domain specifications and logical combinations of these suitable for use in functions such as Reduce and Solve, etc.
- There are several sources for singularities, including Laurent series representation, multivalued functions, and piecewise and partial definitions of functions.
- Singularities from the Laurent series representation where is the location of the isolated singularity:
removable singularity for , e.g. for pole singularity for , e.g. for essential singularity for infinitely many , e.g. for inessential singularity a pole or removable singularity
- Singularities coming from the selection of principal branches of multivalued functions:
branch point point where branches of a multivalued function come together, e.g. for branch cut curve along which a function is discontinuous in order to get a single valued function, e.g. for
- Singularities coming from piecewise-defined functions or natural domain of definition:
piecewise piecewise defined function, e.g. for domain of definition complement of domain of definition, e.g. for
- For a multivariate function, the singularities are taken to be the singularities for each variable separately.
- Possible values for dom are Reals and Complexes.
Examplesopen allclose all
Basic Examples (4)
Basic Applications (5)
Properties & Relations (3)
The function is analytic outside the set given by FunctionSingularities:
Use FunctionAnalytic to check the analyticity:
FunctionDiscontinuities gives a set outside which the function is continuous:
FunctionSingularities finds a condition satisfied by all singularities:
Use SolveValues to find the singularities:
Use FunctionPoles to find the pole singularities and their multiplicities:
Wolfram Research (2020), FunctionSingularities, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionSingularities.html.
Wolfram Language. 2020. "FunctionSingularities." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionSingularities.html.
Wolfram Language. (2020). FunctionSingularities. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionSingularities.html