FunctionSingularities

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 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.

Examples

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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 :

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 neighborhood of :

Check that is analytic at :

Compute the series of at to order : approximates well near :

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:

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

FunctionSingularities finds a condition satisfied by all singularities:

Use SolveValues to find the singularities:

Use FunctionPoles to find the pole singularities and their multiplicities:

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: 