mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except...

poles, that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at...

{\displaystyle \lim _{s\to 1}(s-1)\zeta (s)=1.} Thus the Riemann zeta function is a meromorphic function on the whole complex plane, which is holomorphic everywhere...

contrast to the term meromorphic derived from μέρος (méros) meaning "part". A holomorphic function resembles an entire function ("whole") in a domain...

concerns the existence of meromorphic functions with prescribed poles. Conversely, it can be used to express any meromorphic function as a sum of partial fractions...

through the north pole (that is, the point at infinity). A meromorphic function is a complex function that is holomorphic and therefore analytic everywhere...

The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic...

In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An L-series...

everywhere except a set of isolated points are known as meromorphic functions. On the other hand, the functions z ↦ ℜ ( z ) {\displaystyle z\mapsto \Re (z)} ,...

Instead, modular functions are meromorphic: they are holomorphic on the complement of a set of isolated points, which are poles of the function. A modular form...

field. Patching the local data of meromorphic functions, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the...

meromorphic functions. The function field of a variety is then the set of all meromorphic functions on the variety. (Like all meromorphic functions,...

elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they...

role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic. A ℘-function together with its derivative can be used to parameterize...

analytically to an entire function. A transcendental entire function is an entire function that is not a polynomial. Just as meromorphic functions can be viewed as...

complex rational function with degree one is a Möbius transformation. Rational functions are representative examples of meromorphic functions. Iteration of...

zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative. If f is a meromorphic function inside and on some...

the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is a meromorphic function on the complex plane...

of complex analysis, Nevanlinna theory is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl called...

Weierstrass function ℘τ(z) belonging to the lattice Z + τZ is a meromorphic function on T. This function and its derivative ℘τ′(z) generate the function field...

itself be a meromorphic doubly periodic function with just one zero. Elliptic function Abel elliptic functions Jacobi elliptic functions Weierstrass elliptic...

questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data. They were introduced...

satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its...

integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function f : C ∖ {...

In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers C {\displaystyle \mathbb {C} } defined as the (m +...

associated entire function with zeroes at precisely the points of that sequence. A generalization of the theorem extends it to meromorphic functions and allows...

holomorphic function, then a {\displaystyle a} is an isolated singularity of f {\displaystyle f} . Every singularity of a meromorphic function on an open...

geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex...

derivative exists in this more general region, making the zeta function a meromorphic function. The above equation no longer applies for these extended values...

distributions. The Laplace transform of the Heaviside step function is a meromorphic function. Using the unilateral Laplace transform we have: H ^ ( s )...