test whether is a meromorphic function of x.
test whether is a meromorphic function of x1,x2,….
test whether are meromorphic functions for x1,x2,….
test whether are meromorphic functions for xvars in an open set containing the solutions of the constraints cons.
Details and Options
- A function is meromorphic if it can be represented as , where and are complex analytic functions.
- A function is meromorphic if it can be locally represented as , where and are complex analytic functions.
- If funs contains parameters other than xvars, the result is typically a ConditionalExpression.
- cons can contain inequalities or logical combinations of these.
- The following options can be given:
Assumptions $Assumptions assumptions on parameters GenerateConditions True whether to generate conditions on parameters PerformanceGoal $PerformanceGoal whether to prioritize speed or quality
- Possible settings for GenerateConditions include:
Automatic nongeneric conditions only True all conditions False no conditions None return unevaluated if conditions are needed
- Possible settings for PerformanceGoal are "Speed" and "Quality".
Examplesopen allclose all
Basic Examples (3)
By default, FunctionMeromorphic may generate conditions on symbolic parameters:
Use PerformanceGoal to avoid potentially expensive computations:
Classes of Meromorphic Functions (7)
Functions with branch cuts like Log are not meromorphic:
Neither are Sqrt or any noninteger power:
As all trigonometric and hyperbolic functions are arithmetic combinations of Exp, they are all meromorphic:
Visualize Exp and the eight nonanalytic trigonometric and hyperbolic functions:
The complete Beta function is meromorphic:
It can be considered a multivariate rational function in Gamma:
Integrating Functions (5)
Sqrt has points where the limit does not exist, so it cannot be meromorphic:
The singular points of a meromorphic function, called poles, have a Residue associated with them:
The integral of a meromorphic function around a closed contour equals times the sum of the residues of the poles enclosed by by the curve. Compute the integral of around the origin, which is clearly its only pole:
If all the singular points of a function have the same or related residues, integrals over a closed contour can be used to count the number of poles enclosed. For example, has a pole with residue at every half-integer multiple of :
A common application of contour integrals is to evaluate integrals over the real line, by extending the contour to a closed one with a semicircle in the upper or lower half-plane. If the portion of the integral over the semicircle vanishes, the contour integral must equal the real integral. Consider . The integrand is meromorphic:
For integrands of the form with , meromorphic, continuous on , and for large , the integral can be computed as times the sum of the residues of in the upper half-plane. Use this to compute . First, verify that is meromorphic:
Properties & Relations (5)
Use D to compute derivatives:
Use Series to compute initial terms of the Taylor series:
Use Solve to find the zeros of in the unit disk:
Use FunctionSingularities to find the poles of in the unit disk:
The argument principle states that the difference between the number of zeros and the number of poles of (counted with multiplicities) is given by . Use NIntegrate to compute :
Use Limit to verify that all are simple poles of :
Use FunctionAnalytic to check that and are analytic:
Use FunctionSingularities to find a condition satisfied by all singularities:
Use SolveValues to find the singularities:
Use FunctionPoles to find the poles and their multiplicities:
Use ResidueSum to compute the sum of residues in the right half-plane:
Wolfram Research (2020), FunctionMeromorphic, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionMeromorphic.html.
Wolfram Language. 2020. "FunctionMeromorphic." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionMeromorphic.html.
Wolfram Language. (2020). FunctionMeromorphic. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FunctionMeromorphic.html