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...

\Re (z)>0\,.} The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in...

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...

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...

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...

{\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...

Rational functions are representative examples of meromorphic functions. Iteration of rational functions (maps) on the Riemann sphere creates discrete dynamical...

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

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...

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...

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

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

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...

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...

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

Weierstrass function ℘ τ ( z ) {\displaystyle \wp _{\tau }(z)} belonging to the lattice Z + τ Z is a meromorphic function on T. This function and its derivative...

mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable...

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

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...

= f ( z ) g ( z ) {\displaystyle h(z)={\frac {f(z)}{g(z)}}} is a meromorphic function and a {\displaystyle a} is its pole closest to the origin with order...

definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where...

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

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

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

function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry...

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

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

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

mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic...