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Topic: Algebraic functions

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	Springer Online Reference Works
		The theory of algebraic functions was studied in the past from three different points of view: the function-theoretical point of view taken, in particular, by N.H. Abel, K.
		In the first case the algebraic function is a rational, in the second case it is an elliptic, while in the third case it is a general function.
		Each algebraic function field in one variable is the field of fractions of a Dedekind ring, so that many results and concepts of the theory of divisibility in algebraic number fields can be applied to function fields [12].
	eom.springer.de /A/a011490.htm (1619 words)

	PlanetMath: algebraic function
		A function of one variable is said to be algebraic if it satisfies a polynomial equation whose coefficients are polynomials in the same variable.
		Many special functions, such as Bessel functions, elliptic integrals, and others are known to be transcendental.
		This is version 3 of algebraic function, born on 2005-05-31, modified 2005-05-31.
	planetmath.org /encyclopedia/AlgebraicFunction.html (112 words)

	[No title]
		For algebraic functions, this means the viewing window should be just large enough to display all x- and y-intercepts, all maxima and minima, all points of inflections, all holes, all 2-hole jumps, all vertical asymptotes, all horizontal asymptotes, all vertical tangents, all vertical cusps, all corners, and all isolated points.
		For algebraic functions, vertical asymptotes always occur at the zeros of the denominator that are not eliminated by improper cancellation.
		When an algebraic function has a common factor in the numerator and denominator that can be eliminated in the denominator by improper cancellation, the importance of this lies in the fact that the algebraic function is not defined at the zero of the common factor since it appears in the denominator.
	www.webgraphing.com /algebraicdefs.jsp (2208 words)

	Algebraic curve - Wikipedia, the free encyclopedia
		In algebraic geometry, an algebraic curve is an algebraic variety of dimension one.
		There is a triple equivalence of categories between the category of irreducible projective algebraic curves over the complex numbers, the category of compact Riemann surfaces, and the category of complex algebraic function fields, so that in studying these subjects we are in a sense studying the same thing.
		For this definition to be correct, we must use an algebraically closed field and a curve C in projective space (i.e., complete in the sense of algebraic geometry).
	en.wikipedia.org /wiki/Algebraic_curve (1914 words)

	4.1 - Exponential Functions and Their Graphs
		Algebraic functions are functions which can be expressed using arithmetic operations and whose values are either rational or a root of a rational number.
		Transcendental functions return values which may not be expressible as rational numbers or roots of rational numbers.
		However when transcendental and algebraic functions are mixed in an equation, graphical or numerical techniques are sometimes the only way to find the solution.
	www.richland.edu /james/lecture/m116/logs/exponential.html (1023 words)

	Just WHAT IS ALGEBRAIC THINKING
		The line between the study of informal algebraic ideas and formal algebra is often blurred, and the algebra ideas identified here are intended to be studied in concrete or familiar contexts so that the student will develop a strong conceptual foundation for later in-depth and abstract study of mathematics.
		In this framework, algebraic ideas are viewed in three ways: algebra as abstract arithmetic, algebra as a language, and algebra as a tool for the study of functions and mathematical modeling.
		Algebra is sometimes referred to as generalized or abstract arithmetic, and exploring properties of numbers while developing number sense and operation sense in the elementary years can lay a solid foundation for its formal development.
	www.math.ucla.edu /~kriegler/pub/algebrat.html (2313 words)

	Trigonometry and Basic Functions - Numericana
		Polynomial functions: The value y is obtained from the variable x using only a finite number of additions and/or multiplications involving given constants.
		Algebraic functions: The term applies to any function for which the value y and the variable x are algebraically related, which is to say that there's a two-variable polynomial P such that
		Alternately, such functions may be construed as univalued (ordinary) functions of a variable whose domain is a so-called Riemann surface for which several points may have the same projection on the complex plane.
	home.att.net /~numericana/answer/functions.htm (3889 words)

	PlanetMath: algebraic manifold
		It can be proved that such a manifold is defined as the zero set of a finite collection of analytic algebraic functions.
		Cross-references: analytic algebraic functions, collection, finite, zero set, manifold, variety, algebraic, irreducible, submanifold, field
		This is version 2 of algebraic manifold, born on 2005-12-05, modified 2005-12-05.
	planetmath.org /encyclopedia/KAlgebraicManifold.html (101 words)

	Algebraic Function Fields [HB 57]
		The functionality of extensions of algebraic function fields has been extended to almost match that of ordinary algebraic function fields.
		The functionality of the orders of algebraic function fields has been extended to almost match the functionality of the orders of algebraic number fields.
		All functions for algebraic function fields can be called on extensions of function fields with the exception of those involving series rings, galois groups and subfields.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/rel/node31.htm (533 words)

	[No title]
		For example, sin(x) is non-constant periodic, while the only periodic algebraic functions are constants (this has to be proved), so sin(x), while being analytic, is transcendental.
		I believe the class of elementary functions is a proper subset of the collection of functions described in the next sentence.
		A larger class of functions are those functions that are solutions to a linear differential equation with rational function coefficients.
	www.math.niu.edu /~rusin/known-math/00_incoming/transc_func (1327 words)

	Algebraic variety - Wikipedia, the free encyclopedia
		Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) algebraic geometry.
		Historically, the fundamental theorem of algebra established a link between algebra and geometry by saying that a polynomial in one variable over the complex numbers is determined by the set of its roots, which is an inherently geometric object.
		An abstract algebraic variety is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings.
	en.wikipedia.org /wiki/Algebraic_variety (1098 words)

	Algebraic Operations on Functions (Site not responding. Last check: 2007-11-03)
		In calculus we work with functions whose domains and codomains (ranges) are subsets of the real numbers.
		For an applicaton, see the discussion of algebraic functions in the section on general types of functions.
		Given two functions f(x) and g(x) whose domains are ranges are subsets of the real numbers, we define (f + g)(x) to be the function whose value at x is the sum of f(x) and g(x).
	oregonstate.edu /instruct/mth251/cq/FieldGuide/algebra/lesson.html (197 words)

	Ch. 111, TEKS for Mathematics. Subchapter C. High School
		The study of functions, equations, and their relationship is central to all of mathematics.
		Students perceive functions and equations as means for analyzing and understanding a broad variety of relationships and as a useful tool for expressing generalizations.
		Algebra Though Modeling with the TI-82 and TI-83 Graphing Calculators - an interactive tutorial that starts from scratch with turning on the TI-82 calculator and takes you through the basic steps needed to do arithmetic and function evaluation and to enter, graph, and tabulate functions.
	www.tenet.edu /teks/web/math/alg2.html (2538 words)

	The Math Forum - Math Library - Functions
		An illustrated introductions to the concept of functions, a relationship or correlation where each element of the domain corresponds to one and only one element of the range.
		A college algebra course to provide students with an understanding of the general concepts of relation and function and specifically of polynomial, exponential, and logarithmic functions; with the ability to solve system of equations by utilizing matrices...more>>
		Elementary functions open new teaching and learning possibilities for dynamic geometry, which is characterized particularly by the direct manipulation and production of graphic objects.
	mathforum.org /library/topics/functions (2108 words)

	Hamilton's theory of Conjugate Functions, or Algebraic Couples (Site not responding. Last check: 2007-11-03)
		William R. Hamilton developed a theory of conjugate functions, or algebraic couples, in which complex numbers are constructed as ordered pairs of real numbers.
		On conjugate functions, or algebraic couples, as tending to illustrate generally the doctrine of imaginary quantities, and as confirming the results of Mr Graves respecting the existence of two independent Integers in the complete expression of an imaginary logarithm
		Theory of conjugate functions, or algebraic couples; with a preliminary and elementary essay on algebra as the science of pure time
	www.maths.tcd.ie /pub/HistMath/People/Hamilton/Couples.html (216 words)

	Algebraic Functions (Site not responding. Last check: 2007-11-03)
		Calls the std::exp() function on each element in the matrix.
		Calls the std::log() function on each element in the matrix.
		Calls the std::log1p() function on each element in the matrix.
	www.psatellite.com /products/html/matrixlib_api/a00019.html (806 words)

	L-functions from algebraic geometry (Site not responding. Last check: 2007-11-03)
		Both the Riemann zeta function and the Hasse-Weil L-functions are examples of L-functions, and their occurrence on the Clay list illustrates the central role that L-functions play in number theory in general and in arithmetic algebraic geometry in particular.
		`Arithmetic algebraic geometry' is the modern name for the age-old theory of diophantine equations, with an emphasis on the use of tools from algebraic geometry.
		The aim of the workshop is to explore general arithmetic and analytic properties of L-functions that arise from algebraic varieties.
	www.math.leidenuniv.nl /~psh/wan.shtml (395 words)

	Algebraic Functions (Site not responding. Last check: 2007-11-03)
		Such a combination of integer power functions is called a polynomial.
		Finally, we will examine the inverses of the power functions, known as radical functions.
		All of these functions are said to be algebraic since (unlike the exponential, logarithmic, and trigonometric function) they can be defined by algebraic formulas.
	campus.northpark.edu /math/PreCalculus/Algebraic/index.html (109 words)

	Adding Algebraic Functions (Site not responding. Last check: 2007-11-03)
		Knowledge of algebraic functions and equations is necessary.
		The students should have some knowledge of how to add algebraic functions and be able to recognize that "x" terms can not be added to whole numbers(2).
		The idea of the lesson should be explained stating that we are going to experiment with using calculators to show graphs of algebraic functions.
	www.edb.utexas.edu /faculty/wstroup/Gen_Act_Web/G36/zga36.html (583 words)

	Amazon.com: Number Theory: Algebraic Numbers and Functions (Graduate Studies in Mathematics): Books: Helmut Koch,David ... (Site not responding. Last check: 2007-11-03)
		Algebraic number theory is one of the most refined creations in mathematics.
		The primary goal of this book is to present the essential elements of algebraic number theory, including the theory of normal extensions up through a glimpse of class field theory.
		This is done on the one hand to demonstrate the analogy between number fields and function fields, which is especially clear in the case where the ground field is a finite field.
	amazon.com /Number-Theory-Algebraic-Functions-Mathematics/dp/0821820540 (735 words)

	Algebraic Number Theory Archive (Site not responding. Last check: 2007-11-03)
		math.NT/0408383: 27 Aug 2004, Zeta functions of supersingular curves of genus 2, by Daniel Maisner, Enric Nart.
		math.NT/0310162: 11 Oct 2003, Algebraic cycles on Hilbert modular fourfolds and poles of L-functions, by Dinakar Ramakrishnan.
		ANT-0188: 15 Jun 1999, The Mordell-Weil rank of the Jacobian of a curve of genus 2 with multiplication by a square root of 2, by Peter R. Bending.
	front.math.ucdavis.edu /ANT (12251 words)

	Amazon.com: Complex Functions: An Algebraic and Geometric Viewpoint: Books: Gareth A. Jones,David Singerman (Site not responding. Last check: 2007-11-03)
		Algebraic Curves and Riemann Surfaces (Graduate Studies in Mathematics, Vol 5) (Graduate Studies in Mathematics, Vol 5) by Rick Miranda
		At the present time there is a great revival of interest in these topics not only for their own sake but also because of their applications to so many areas of mathematical research from group theory and number theory to topology and differential equations.
		There are several advantages in using the set C of complex numbers as the domain of definition of functions.
	www.amazon.com /Complex-Functions-Algebraic-Geometric-Viewpoint/dp/052131366X (1233 words)

	Algebraic Functions
		A function is a rule which indicates an operation to perform.
		The inverse function of y = 2x is y = ½x.
		The inverse of a function does the opposite of the function.
	www.revision-notes.co.uk /revision/734.html (314 words)

	Natural logarithm: Representations through more general functions
		Classical cases involving algebraic functions in the arguments and unit step theta (4 formulas)
		Generalized cases for powers of log involving algebraic functions in the arguments (8 formulas)
		Generalized cases involving algebraic functions in the arguments and unit step theta (4 formulas)
	functions.wolfram.com /ElementaryFunctions/Log/26/ShowAll.html (143 words)

	NSDL Metadata Record -- Algebraic Function -- from MathWorld
		A function which can be constructed using only a finite number of elementary operations together with the inverses of functions capable of being so constructed.
		Flajolet, P. and Sedgewick, R. "Analytic Combinatorics: Functional Equations, Rational and Algebraic Functions." http://www.inria.fr/RRRT/RR-4103.html.
		Koch, H. "Algebraic Functions of One Variable." Ch.
	nsdl.org /mr/697686 (107 words)

	Math Function Mania : Learn algebraic functions with this game.
		Math Function Mania is a fun multimedia game that teaches functions, algebra and problem solving skills.
		In the "Mania" game you must first detect which function is being used, and then solve it by clicking on the correct multiple choice answer.
		Topics covered include equations, algebra, problem solving, critical thinking, polynomials, factoring, remainders, number bases, and prime numbers.
	fileprofile.com /Education/mathematics/math_function_mania (283 words)

	What's New in Mathematica 5: Other Functions: Algebraic Number Objects
		object contains the minimal polynomial of the algebraic number and the root number--an integer indicating which of the roots of the minimal polynomial the
		Within a fixed algebraic number field, the algebraic number arithmetic in the
		common algebraic number field containing {a, b, c}, which takes only a fraction of a second.
	www.wolfram.com /products/mathematica/newin5/numberobjects.html (166 words)

	Algebraic Functions (via CobWeb/3.1 planetlab-3.cs.princeton.edu) (Site not responding. Last check: 2007-11-03)
		This appendix lists the functions which can be used in algebraic expressions (see Section 6).
		They are listed in two sections: the first gives the functions available for use anywhere an expression can be used, and the second gives those only for use in defining custom Activation Actions.
		Note that although all the available functions are listed here with short descriptions, their full explanation, including parameter descriptions and examples, is only available from the Available Functions Window, obtained using the
	www.star.bris.ac.uk.cob-web.org:8888 /~mbt/topcat/sun253/funcs.html (89 words)

	Algebraic Function -- from Wolfram MathWorld
		A function which can be constructed using only a finite number of elementary operations together with the
		inverses of functions capable of being so constructed.
		SEE ALSO: Elementary Function, Elementary Operation, Transcendental Function.
	mathworld.wolfram.com /AlgebraicFunction.html (116 words)

	Algebraic Functions and Projective Curves (via CobWeb/3.1 planetlab-3.cs.princeton.edu) (Site not responding. Last check: 2007-11-03)
		Algebraic Functions and Projective Curves (via CobWeb/3.1 planetlab-3.cs.princeton.edu)
		This site is the home page of Algebraic Functions and Projective Curves, published by Springer-Verlag The book is available online at Amazon.com.
		Subscribe to the newletter to receive announcements about the book such as new printings or editions.
	www.functionfields.org.cob-web.org:8888 (203 words)

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