FunctionMeromorphic cannot find the answer for arbitrary values of the parameter :

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

By default, FunctionMeromorphic may generate conditions on symbolic parameters:

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

This returns a conditionally valid result without stating the condition:

By default, all conditions are reported:

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

Use PerformanceGoal to avoid potentially expensive computations:

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

Rational functions are meromorphic:

Tan, Sec and Sech are meromorphic:

Visualize these functions:

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

Functions with branch cuts like Log are not meromorphic:

Neither are Sqrt or any noninteger power:

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

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

Visualize some of these functions:

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

These functions are not defined for complex value:

Every function analytic in the complex plane is meromorphic:

Arithmetic combinations of meromorphic functions are meromorphic:

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

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

Visualize Exp and the eight nonanalytic trigonometric and hyperbolic functions:

The compositions of meromorphic functions need not be meromorphic:

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

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

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

Visualize the composite functions:

Multivariate rational functions are meromorphic:

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

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

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

The complete Beta function is meromorphic:

It can be considered a multivariate rational function in Gamma:

Visualize the function:

The Limit of a meromorphic function is always either a number or ComplexInfinity:

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

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:

This must equal the residue at :

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

Visualize the function and the contours:

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

Visualize the function with this alternate contour:

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 :

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

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:

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

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

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

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:

The function also decays at infinity as is required:

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

Thus, the integral :

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

Verify the result with a direct computation: