FunctionSingularities

FunctionSingularities[f,x]

finds the singularities of for xReals.

FunctionSingularities[f,x,dom]

finds the singularities of for xdom.

FunctionSingularities[{f1,f2,},{x1,x2,},dom]

finds the singularities of for x1,x2,dom.

Details

  • 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 singularitya pole or removable singularity
  • Singularities coming from the selection of principal branches of multivalued functions:
  • branch pointpoint where branches of a multivalued function come together, e.g. for
    branch cutcurve 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:
  • piecewisepiecewise defined function, e.g. for
    domain of definitioncomplement of domain of definition, e.g. TemplateBox[{1, {1, /, 2}, x}, EllipticTheta] for TemplateBox[{x}, Abs]>=1
  • For a multivariate function, the singularities are taken to be the singularities for each variable separately.
  • Possible values for dom are Reals and Complexes.

Examples

open allclose all

Basic Examples  (4)

Find the singularities of a real univariate function:

Find the singularities of a complex univariate function:

Find the singularities of a real multivariate function:

Find the singularities of a complex multivariate function:

Scope  (5)

Singularities of a real univariate function:

Find the singular points between and :

Visualize the singularities:

Singularities of a function composition:

Find the singular points between and :

Visualize the singularities:

Singularities over the reals include the points where the function is not real valued:

Singularities of a complex univariate function:

Compute the singularities in terms of Re[z] and Im[z]:

Visualize the singularities:

Singularities of a real multivariate function:

Visualize the singularities:

Applications  (6)

Basic Applications  (5)

Find the singularities of :

Find the singular points between and :

Visualize the singularities:

Find the singularities of max(log(TemplateBox[{x}, Abs]+1),x sin(x)):

Find the singular points between and :

The function is continuous but not analytic:

Find the singularities of :

Show that there are no singularities:

The function is analytic:

Find the singularities of the complex function :

Compute the singularities in terms of Re[z] and Im[z]:

Visualize the singularities:

Find the singularities of given the singularities of and :

Suppose the singularities of and are contained in solution sets of and :

The singularities of are contained in the solution set of :

Calculus  (1)

If is analytic at , then is the limit of its Taylor series in a neighbourhood of :

Check that is analytic at :

Compute the series of at to order :

approximates well near :

Properties & Relations  (2)

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:

The set of discontinuities is a subset of the set of singularities:

Possible Issues  (2)

The singularity set returned may not be minimal:

The function is identically zero, hence it has no singularities:

When some singularity information is missing, an error message is given and the known singularities are returned:

Wolfram Research (2020), FunctionSingularities, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionSingularities.html.

Text

Wolfram Research (2020), FunctionSingularities, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionSingularities.html.

BibTeX

@misc{reference.wolfram_2021_functionsingularities, author="Wolfram Research", title="{FunctionSingularities}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionSingularities.html}", note=[Accessed: 19-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_functionsingularities, organization={Wolfram Research}, title={FunctionSingularities}, year={2020}, url={https://reference.wolfram.com/language/ref/FunctionSingularities.html}, note=[Accessed: 19-October-2021 ]}

CMS

Wolfram Language. 2020. "FunctionSingularities." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionSingularities.html.

APA

Wolfram Language. (2020). FunctionSingularities. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionSingularities.html