
	Where results make sense

Topic: Meromorphic function

Related Topics

function mathematics

functional analysis

functional programming

trigonometric function

functional group

exponential function

probability density function

continuous functions

graph of a function

Riemann zeta function

function domain

holomorphic function

In the News (Fri 17 Feb 12)

EXECUTIVE POWER

Iran

ANALYSIS

Pakistan

Family compact

	Meromorphic function - Wikipedia, the free encyclopedia
		In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function.
		Every meromorphic function on D can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D: the poles then occur at the zeroes of the denominator.
		Thus, this function is not meromorphic in the whole complex plane.
	en.wikipedia.org /wiki/Meromorphic_function (407 words)

	Meromorphic function at opensource encyclopedia (Site not responding. Last check: 2007-11-07)
		A meromorphic function is a function that is holomorphic everywhere on the complex plane except at points in a set of isolated poles, which are certain well-behaved singularities.
		Every meromorphic function can be expressed as the ratio between two entire functions (with the denominator not constant 0): the poles then occur at the zeroes of the denominator.
		In the language of Riemann surfaces, a meromorphic function is the same as a holomorphic function from the complex plane to the Riemann sphere which is not constant ∞.
	www.wiki.tatet.com /Meromorphic.html (206 words)

	Cousin problems - Wikipedia, the free encyclopedia
		In mathematics, the Cousin problems are two questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data.
		The attack on this problem by means of taking logarithms, to reduce it to the additive problem, meets an obstruction in the form of the first Chern class; and this implies that it cannot always be solved on a Stein manifold M unless the cohomology group
		In terms of sheaf theory, these problems can be expressed via quotient sheaves: of the sheaf of meromorphic functions modulo holomorphic functions, for the first problem, and for the sheaf of non-vanishing meromorphic functions modulo non-vanishing holomorphic functions, in the second case.
	en.wikipedia.org /wiki/Cousin_problems (320 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-07)
		A meromorphic function is a complex function that is analytic everywhere except for possibly a discrete subset of its domain, that is, except at some, possibly infinitely many, points.
		Here is the definition given by the CRC Concise Encyclopedia of Mathematics: A meromorphic function is complex analytic in all but a discrete subset of its domain, and at those singularities it must go to infinity like a polynomial (i.e., have no essential singularities).
		An equivalent definition of a meromorphic function is a complex analytic map to the Riemann Sphere.
	mathforum.org /library/drmath/view/53856.html (199 words)

	[No title]
		Given a complex function f on the boundary of the unit circle can you tell when it can be analytically extended inside.
		Suppose you have a holomorphic function in a strip, continuous and bounded in absolute value by 1 on the boundary, and bounded everywhere.
		Elliptic functions ------------------ Talk about doubly periodic functions on C. Prove that the sum of the residues of such a function in a period parallelogram is 0.
	www.princeton.edu /~missouri/Generals/generals/complex.txt (1516 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The field of meromorphic functions on $T$ can be identified with the field of doubly-periodic functions on $\C$, which in turn can be generated by the Weierstra\ss\ $\wp$ function and its derivative $\wp'$ subject to the relation $$\wp'(z,\tau)^2=4\wp(z,\tau)^3-g_2(\tau)\wp(z,\tau)-g_3(\tau)\.\tag 3-1$$ Here everything depends on the lattice parameter $\tau $.
		The field of meromorphic function on the torus corresponds to the field of rational functions on the curve $E$ $$\C(X)[Y]/\big(Y^2-f(X)\big), \quad f(T)=4(T-e_1)(T-e_2)(T-e_3)\.\tag 3-7$$ Every meromorphic function on the torus can be described as a rational function (i.e.
		A function $f$ is called regular on a subset of the curve $E$ if $f$ can be given as quotient of polynomials such that the the denominator polynomial does not vanish on $E$.
	www.ma.utexas.edu /mp_arc/papers/92-12 (6353 words)

	PlanetMath: gamma function (Site not responding. Last check: 2007-11-07)
		Some values of the gamma function for small arguments are:
		The gamma function has a meromorphic continuation to the entire complex plane with poles at the non-positive integers.
		This is version 11 of gamma function, born on 2001-11-17, modified 2005-01-12.
	planetmath.org /encyclopedia/GammaFunction.html (119 words)

	[No title]
		Assuming that the size of a generation of a population depends solely on the size of the previous generation and may thus be expressed as a function of it, questions concerning the further development of the population reduce to iteration of this function.
		From this point of view, the iteration theory of entire functions and of meromorphic functions with one pole which is an omitted value is quite different from that of general meromorphic functions, which have at least two poles or only one pole which is not omitted.
		For transcendental functions, $\Si(f^{-1})$ may of course be infinite (and simple examples like $f(z)=e^z+z+1$ or $f(z)=e^z+z+2$ show that there may, in fact, be infinitely many cycles of immediate attractive basins and Leau domains), but for a rational function $f$ of degree $d$ there are at most $2d-2$ singularities of $f^{-1}$.
	www.ams.org /journals/bull/pre-1996-data/199329-2/Bergweiler (9297 words)

	ass7 (Site not responding. Last check: 2007-11-07)
		Recall that we proved that a meromorphic function on the sphere was a rational function, and hence, the sum of the orders of the zeros (including those at oo) equals the sum of the orders of the poles (including those at oo).
		However, it is valid for a meromorphic 1 form rather than a meromorphic function.
		For any nonconstant meromorphic function f on a compact Riemann surface, the sum of the orders of the zeros of f equals the sum of the orders of the poles of f.
	math.rice.edu /~hardt/428/ass7.html (186 words)

	Elliptic Curves and Modular Functions
		Modular and elliptic functions are both special cases of the concept of an automorphic function, which is a meromorphic function of 1 or more complex variables defined on a particular complex manifold and invariant under a particular group of analytic transformations (symmetries) of the manifold.
		the modular functions, are essentially the meromorphic functions on D considered as a Riemann surface by its isomorphism with H/ We have stressed this idea of symmetry because of the way it relates the analytic and geometric properties of an object like a Riemann surface to the algebraic properties of a group.
		Thus the elliptic functions are essentially the automorphic functions on the extended complex plane corresponding to the group of translations by two non-collinear values.
	www.mbay.net /~cgd/flt/flt05.htm (2994 words)

	Argument Principle and Rouche's Theorem
		They have important practical applications and pertain only to functions all of whose isolated singularities are poles.
		Analytic functions are a special case of meromorphic functions.
		Theorem 8.8, known as the argument principle, is useful in determining the number of zeros and poles that a function has.
	math.fullerton.edu /mathews/c2003/RoucheTheoremMod.html (560 words)

	L-functions and elliptic curves
		For many such series of interest, the corresponding function can be extended to a meromorphic function on the whole plane by a process known as "analytic continuation".
		This function is known as the Hasse-Weil L-function.
		Since this result demonstrates the functional equation for L(E,s), it verifies the Hasse-Weil conjecture when E is a modular curve (i.
	www.mbay.net /~cgd/flt/flt06.htm (2077 words)

	HKUST Institutional Repository: Item 1783.1/1563 (Site not responding. Last check: 2007-11-07)
		Finally, we study the zeros of derivatives and homogeneous differential polynomials of meromorphic functions In Chapter 2, we will first give a normal criterion relating to algebraic differential equations and discuss the growth of the specially entire solutions of some algebraic differential equations.
		Then the growth of the meromorphic solutions of the equation f ''= L(z, f)(f ')2 + M(z, f)f '+ N(z, f), where L, M, N are birational functions, is studied.
		We will give an estimation of the Nevanlinna characteristic function of Riemann zeta function, then by using this, we will prove that Riemann zeta function is prime.
	hdl.handle.net /1783.1/1563 (418 words)

	Holomorphic and Meromorphic Functions (Site not responding. Last check: 2007-11-07)
		A holomorphic function on the plane (or other complex manifold) is analytic.
		A meromorphic function is the quotient of two holomorphic functions f/g.
		Technically, every holomorphic function is meromorphic, since you can always set g to 1.
	www.mathreference.com /cx-pow,holom.html (149 words)

	Weierstrass P-Function
		A singularly periodic function p(x) on R which satisfies f(x + nw) = f(x) is determined by the values on the interval [0,w] - the values at 0 and w are the same, so it can be pictured as two endpoints that have been glued together.
		By Liouville's theorem, a meromorphic function which is bounded on all of C must be a constant.
		To set the stage for an important fact, since a meromorphic function (analytic in nature, except at a discrete set of points were it has singularity - or poles - everywhere else it is differentiable) can only have finitely many poles.
	www.willamette.edu /~zizza/Courses/SeniorSeminar/G2.3/KD3.html (1290 words)

	No Title
		Lang, page 171 Problem 10: Show that any function, which is meromorphic on the extended complex plane, is a rational function.
		be an isolated singularity of an analytic function f.
		A function f is said to be doubly periodic with periods 1 and
	www.math.umass.edu /~markman/m621_spring00_html/hw5/hw5.html (543 words)

	[No title]
		Lattices in (and the Field of Elliptic Functions EL A lattice L in the complex plane is the set of all integral linear combinations of two complex numbers (1 and (2, where (1 and (2 are linearly independent.
		For a given lattice L, a meromorphic function on (is an elliptic function iff it is doubly periodic.
		We define the Weierstrass function ((z) with respect to a normalized lattice L in the complex plane as follows: EMBED Equation.3 for l (L. It can be shown that: ((z) converges uniformly and absolutely on compact subsets of (/L; and ((z) (EL, and its only poles are double poles on its lattice points.
	www.ms.uky.edu /~uwenagel/ALG-GEOM-04/watson.doc (1428 words)

	AMCA: Hurwitz spaces by Sergei Natanzon
		is a set of all analytic meromorphic function of genus g and degree n.
		be the Hurwitz space of all meromorphic functions of a topological type t.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/c/y/10.htm (421 words)

	PlanetMath: $\wp$-function (Site not responding. Last check: 2007-11-07)
		is also an odd, meromorphic, and elliptic function, analytic at
		form together a generator set for the field of elliptic functions associated to the lattice
		Cross-references: field, generator, elliptic function, odd, derivative, even function, pole, order, analytic, meromorphic, function, lattice
	planetmath.org /encyclopedia/WeierstrassWpFunction.html (77 words)

	On the Singularities of the Inverse to a Meromorphic Function of Finite Order - Bergweiler, Eremenko (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		For example we prove that if f is a transcendental meromorphic function then f # f n with n # 1 takes every finite non-zero value infinitely often.
		28 Iteration of meromorphic functions (context) - Bergweiler - 1993
		2 The set of asymptotic values of a meromorphic function of fi..
	citeseer.ist.psu.edu /408043.html (353 words)

	140a
		is (conjecturally) not continuable to a meromorphic function, and in fact it has a natural boundary.
		Diaconu, Goldfeld, and Hoffstein have shown that the function continues to a sufficiently large region that standard conjectures for the moments of
		However, the fact that the function under consideration is not entire suggests that it may not be the correct object to study.
	www.aimath.org /WWN/lrmt/articles/html/140a (231 words)

	HKUST Institutional Repository: Item 1783.1/1507
		Let f(z) be a transcendental function meromorphic in the complex plane.
		We say that a meromorphic function γ(z) is a small meromorphic function with respect to f(z), provided that T(r,γ) = S(r,f).
		Let g(z) be a transcendental entire function, or else meromorphic if f(z) is a rational function.
	hdl.handle.net /1783.1/1507 (165 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Show that it has a meromorphic continuation to the complex plane.
		I started to give a proof using contour integrals, but they wanted to do it using the functional equation, and started asking me about functional equations.
		Consider a function analytic in the unit disc satisfying f(2z) = f(z)/(1+f(z)^2).
	www.math.princeton.edu /graduate/generals/vatsal_vinayak (474 words)

	Math 428: Topics in Complex Analysis, Fall 98
		Weierstrass's work on the analytic continuation of holomorphic functions led Riemann to develop the notion of the Riemann surface as a natural domain for an algebraic function.
		It is well-known that Riemann surface theory is not only a point of departure but even at the historical roots of a large part of mathematics, in particular, topology, differential geometry, algebraic geometry, algebraic number theory, and partial differential equations.
		In the proof of the existence of a nonconstant meromorphic function, we used a holomorphic 1 form
	math.rice.edu /~hardt/428 (1647 words)

	Graphics (Site not responding. Last check: 2007-11-07)
		Picture B. Here's another picture of the Weierstrass P-function, but this time it's regarded as a doubly periodic function on the complex plane (the universal cover of the torus in the previous example).
		The picture-rectangle is a region on the complex plane with the origin at its center.
		Picture C. This depicts a degree-two rational function from the Riemann sphere to itself.
	faculty.rmc.edu /rhammack/pictures.html (507 words)

	Meromorphic function of infinite order with maximum deficiency sum, Weiling Xiong (Site not responding. Last check: 2007-11-07)
		Meromorphic function of infinite order with maximum deficiency sum, Weiling Xiong
		Meromorphic function of infinite order with maximum deficiency sum
		In this paper we prove the following theorem: % Let $f(z)$ be a meromorphic function of infinite order.
	projecteuclid.org /getRecord?id=euclid.kmj/1093351318 (147 words)

	Citebase - The representation of a meromorphic function as the quotient of entire functions and Paley problem in C^n: ...
		The representation of a meromorphic function as the quotient of entire functions and Paley problem in C
		Here we introduce generalizations of the Nevanlinna characteristic and give a short survey of classical and recent results on the representation of a meromorphic function in terms such characteristics.
		When f has a finite lower order, the Paley problem on best possible estimates of the growth of entire functions g and h in the representations f=g/h will be considered.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0502433 (1361 words)

	WeierstrassFactorizationTheorem (Site not responding. Last check: 2007-11-07)
		compact subsets, then it converges to a holomorphic function given that all the
		This is what we will mean by the infinite product in what follows.
		meromorphic function just by dividing two holomorphic functions
	aux.planetmath.org /cache/objects/5794/l2h (221 words)

Try your search on: Qwika (all wikis)