FunctionMeromorphic
✖
FunctionMeromorphic
tests 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".
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
Test whether univariate functions are meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-cn997v


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-htgqql

Test whether multivariate functions are meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-b61dx3


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-drwv62

Test whether functions are meromorphic over restricted domains:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-b3bcbq


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-oz9uup

Scope (4)Survey of the scope of standard use cases

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-lek6

The only singularities are poles:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-bgs0w5


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-hmoak3

It has a branch cut along the negative real axis:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-c2miwu

Functions with restricted domains:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-hv9ld2


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-fl1zj0

The branch cut along the positive imaginary axis is outside the restricted domain:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-cwz8y


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-b2lgms


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-dfhgfg

Functions with symbolic parameters:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-dyuwz0

Options (4)Common values & functionality for each option
Assumptions (1)
FunctionMeromorphic cannot find the answer for arbitrary values of the parameter :

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-bnv0jw

With the assumption that is a positive integer, FunctionMeromorphic succeeds:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-c50qtk

GenerateConditions (2)
By default, FunctionMeromorphic may generate conditions on symbolic parameters:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-osy2z

With GenerateConditions->None, FunctionMeromorphic fails instead of giving a conditional result:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-fe9ubz

This returns a conditionally valid result without stating the condition:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-na5ydu

By default, all conditions are reported:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-tdcquw

With GenerateConditions->Automatic, conditions that are generically true are not reported:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-291b1m

PerformanceGoal (1)
Use PerformanceGoal to avoid potentially expensive computations:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-i86kxj

The default setting uses all available techniques to try to produce a result:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-i9gq

Applications (12)Sample problems that can be solved with this function
Classes of Meromorphic Functions (7)
Rational functions are meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-su2zqy

Tan, Sec and Sech are meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-rz8o0e


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-rz2q46

Visualizing the functions in a plane shows that their singularities are no worse than poles:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-6vtxse

Functions with branch cuts like Log are not meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-oq2mh5

Neither are Sqrt or any noninteger power:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ho2d7w


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-mxi6zp

Inverse trigonometric and hyperbolic functions like ArcSin, ArcTan and ArcCsch are similarly non-meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-c2oecg

Non-differentiable functions like Abs, Sign and Re are not meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ekb094


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-j7h74d

Visualize some of these functions:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-4dqe8k

Functions that are only defined for real inputs, like UnitStep and TriangleWave, cannot be meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-pbgcs0


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-dhm00y

These functions are not defined for complex value:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-g4qojn

Every function analytic in the complex plane is meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-z9zqo5


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-83eftz


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-rr0lvp

Arithmetic combinations of meromorphic functions are meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-0e1wm


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-2zirk2


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-8r4dzm


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-7ty4of

As all trigonometric and hyperbolic functions are arithmetic combinations of Exp, they are all meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-7jfdkz

More generally, any rational combination of meromorphic functions is meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-c0v221


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-oueqxz

Visualize Exp and the eight nonanalytic trigonometric and hyperbolic functions:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ld25ol

The compositions of meromorphic functions need not be meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-el1vuj

The singularities of a meromorphic might bunch up under composition, leading to a non-pole singularity:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-x267y9

There is an essential singularity at the origin, as evidenced by the following limit:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-so5h8v

However, the composition of a meromorphic function with an analytic function is always meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-4brnxz


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-u97pmu


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-oi36v9

Visualize the composite functions:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-2q0zi7

Multivariate rational functions are meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ta9a8e

Unlike functions of a single variable, singularities lie along curves, in the first function along :

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-vscos0

Plotting the second function in the plane shows the blowup along the hyperbolas
:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-9i2g02

By composing with analytic univariate functions, many more analytic functions can be generated:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-pwe370

The complete Beta function is meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-m97fja

It can be considered a multivariate rational function in Gamma:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ib3vsi


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-skyi0r

Integrating Functions (5)
The Limit of a meromorphic function is always either a number or ComplexInfinity:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-j7781e

Sqrt has points where the limit does not exist, so it cannot be meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-l1ja5x

The singular points of a meromorphic function, called poles, have a Residue associated with them:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ywwcbl

The residue is the coefficient of in the power series expansion of the function:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-q6w7jf

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:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-8ohoyf

This must equal the residue at
:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-17wmja

Any other closed contour would give the same result, for example, a circular one:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-6dicla

Visualize the function and the contours:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-zyoxc0

If the contour does not enclose the singularity, the integral will be zero:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-zyr543

Visualize the function with this alternate contour:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-gh4lb9

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
:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-bfk4ae

The integral of over a rectangle straddling the real axis counts the number of half-integer multiples of
enclosed:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-dw7tu6

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:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-dsvevw

The singularities of the integrand occur where the denominator is zero:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-8l3kbl

Complete the contour using a semicircle in the upper half-plane and compute the integral using residues:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-xedo69

The integral over the semicircle is of order as
, so the real integral must have the same value:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-rjk8cw

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:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ylgy5o

The function also decays at infinity as is required:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-8nbwdr

There is a single pole in the upper half-plane at :

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ojwcl0


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-x9yaky

Since and
is even,
is half of the previous result:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-mjuhti

Verify the result with a direct computation:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-evibhy

Properties & Relations (5)Properties of the function, and connections to other functions
A meromorphic function is differentiable arbitrarily many times:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-gi3ncy

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-gn80ue

Use D to compute derivatives:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-dk9bqa

A meromorphic function can be expressed as a Taylor series at each point of its domain:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-cy6veu

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-fc791q

Use Series to compute initial terms of the Taylor series:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-b8crv7

The resulting polynomial approximates near
:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-b997hn

At its poles, the function can be expressed as a Laurent series with a finite principal part:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-iadvj


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-cvdt7a

A meromorphic function can have only finitely many zeros and poles in a compact region:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-fflilk

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ccc4nd

Use Solve to find the zeros of in the unit disk:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-60azb

Use FunctionSingularities to find the poles of in the unit disk:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-csk5jm

Plot , its zeros (blue) and its poles (red):

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-gjv4ax

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
:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ga7dht

Verify that all are simple zeros of
:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-dhmu59

Use Limit to verify that all are simple poles of
:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-feb58


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-e5qew2

A quotient of complex analytic functions is meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-jh7fn8
Use FunctionAnalytic to check that and
are analytic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-dyvh03

Verify that and
are meromorphic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-iszap

Meromorphic functions may not be complex analytic:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ctjm0u


https://wolfram.com/xid/0b7ey6hh6rgh7wyi-jpusg

The only singularities a meromorphic function can have are poles and removable singularities:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-csujlz

Use FunctionSingularities to find a condition satisfied by all singularities:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-bn7hfl

Use SolveValues to find the singularities:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-ehke72

Use FunctionPoles to find the poles and their multiplicities:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-hmn6oq

Use ResidueSum to compute the sum of residues in the right half-plane:

https://wolfram.com/xid/0b7ey6hh6rgh7wyi-c8yjzt

Wolfram Research (2020), FunctionMeromorphic, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionMeromorphic.html.
Text
Wolfram Research (2020), FunctionMeromorphic, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionMeromorphic.html.
Wolfram Research (2020), FunctionMeromorphic, Wolfram Language function, https://reference.wolfram.com/language/ref/FunctionMeromorphic.html.
CMS
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. Wolfram Research. https://reference.wolfram.com/language/ref/FunctionMeromorphic.html.
APA
Wolfram Language. (2020). FunctionMeromorphic. Wolfram Language & System Documentation Center. Retrieved from 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
BibTeX
@misc{reference.wolfram_2025_functionmeromorphic, author="Wolfram Research", title="{FunctionMeromorphic}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/FunctionMeromorphic.html}", note=[Accessed: 08-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_functionmeromorphic, organization={Wolfram Research}, title={FunctionMeromorphic}, year={2020}, url={https://reference.wolfram.com/language/ref/FunctionMeromorphic.html}, note=[Accessed: 08-July-2025
]}