
	Where results make sense

Topic: Elementary function (differential algebra)

Related Topics

abstract algebra

algebraic geometry

linear algebra

algebraic topology

Boolean algebra

Lie algebra

algebraic varieties

basis linear algebra

ring algebra

associative algebra

algebraic notation

homological algebra

In the News (Sat 2 Jun 12)

EXECUTIVE POWER

Iran

ANALYSIS

Pakistan

Family compact

	Differential Galois theory - Wikipedia, the free encyclopedia
		In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions.
		Differential Galois theory is a theory based on the model of Galois theory.
		One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory.
	en.wikipedia.org /wiki/Differential_Galois_theory (586 words)

	Encyclopedia: Equation
		A linear equation in algebra is an equation which is constructed by equating two linear functions.
		Graph of a quadratic function: y = x2 - x - 2 = (x+1)(x-2) The x-coordinatess of the points where the graph crosses the x-axis, x = -1 and x = 2, are the roots of the quadratic equation: x2 - x - 2 = 0.
		In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables.
	www.nationmaster.com /encyclopedia/Equation (582 words)

	The Differential Form
		Trying to analyze a real function as we did for the derivative was complicated by the complexity of the real numbers and the fact that we couldn't move as reliably as we would like from one number to another.
		In analyzing a function for the derivative, we found that we could take the ratio between differences in the value of a function and the differences in the values in the domain and look for some concrete value of the ratio as we neared a selected point.
		When we considered functions on intervals of the real numbers, we examined the values they took on the ends of the interval hoping to recover the use of the difference in determining length as a kind of volume of the interval.
	galileo.spaceports.com /~symbiota/HIR/HIRDifferentDiffForm.htm (7277 words)

	Risch algorithm - Wikipedia, the free encyclopedia
		It is based on the form of the function being integrated and on methods for integrating rational functions, logarithms, and exponential functions.
		The Risch algorithm is used to integrate elementary functions.
		Laplace solved this problem for the case of rational functions, as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions.
	www.wikipedia.org /wiki/Risch_algorithm (391 words)

	Elementary function (differential algebra) - Enpsychlopedia (Site not responding. Last check: 2007-11-04)
		In differential algebra, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ − × ÷).
		The trigonometric functions and their inverses are assumed to be included in the elementary functions by using complex variables (i = √-1) and the relations between the trigonometric functions and the exponential and logarithm functions.
		For polynomials of degree four and smaller there are explicit formulas for the roots (the formulas are elementary functions), but even for higher degree polynomials the fundamental theorem of algebra and the implicit function theorem assures the existence of a function that returns each one of the roots of a polynomial equation.
	www.grohol.com /psypsych/Elementary_functions (537 words)

	Mathematics
		The study of structure starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra.
		The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of functions, fiber bundles, derivatives, smoothness and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations.
		Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions, laying the groundwork for quantum mechanics among many other things.
	www.brainyencyclopedia.com /encyclopedia/m/ma/mathematics.html (2206 words)

	Differential Galois theory -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-04)
		Differential Galois theory is a theory based on the model of (Group theory applied to the solution of algebraic equations) Galois theory.
		Given two differential fields F and G, G is called a logarithmic extension of F if G is a (Click link for more info and facts about simple transcendental extension) simple transcendental extension of F (i.e.
		Suppose F and G are differential fields, with Con(F)=Con(G), and that G an elementary differential extension of F.
	www.absoluteastronomy.com /encyclopedia/D/Di/Differential_Galois_theory.htm (838 words)

	Exponential function - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-04)
		Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.
		The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin.
		Similarly, since the Lie algebra M(n, R) of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.
	www.xahlee.org /_p/wiki/Exponential_function.html (900 words)

	12: Field theory and polynomials
		Algebraic extensions are those in which every element of K is the root of a polynomial with coefficients in F; elements of K which are not algebraic are transcendental over F (e.g.
		Fields of functions of algebraic varieties (essentially the quotient fields of rings F[x1,...,xn]/(P) where P is a multivariable polynomial) are more properly treated in 14: Algebraic Geometry, although these are really just discussions of fields of finite transcendence degree over the ground field.
		Likewise fields of meromorphic functions and local rings of germs of functions are usually treated with their applications to 30: Complex Analysis, 32: Several complex variables, and 58: Analysis on manifolds.
	www.math.niu.edu /~rusin/known-math/index/12-XX.html (1782 words)

	Mathematics of Computer Algebra and Analysis - MOCAA
		All the elementary notions of analysis, such as continuity and differentiability, have to be given precise computational meaning.
		The goal of this project is to extend the database of known special functions in Maple and to develop tools for reliably simplifying formulae containing the elementary functions, special functions, sums and integrals.
		We are developing support for new special functions in Maple, e.g., the family of Heun functions, and new special functions in mathematics, e.g., the Lambert W function.
	www.cecm.sfu.ca /~pborwein/MITACS/Research.htm (4432 words)

	Physics Help and Math Help - Physics Forums - Are certain integrals possible?
		Elementary functions, like log, sin, and arctan already have all their properties very well understood.
		The Fresnel integral is not an elementary function, and not all of its properties are understood yet.
		The theory of which functions have elementary antiderivatives is called differential algebra and I have a book on it but have never read it.
	www.physicsforums.com /printthread.php?t=18831 (1079 words)

	BibTeX bibliography elefunt.bib
		Functions and limits (the foundations of analysis)", title = "Elementary functions of a complex variable", publisher = "Gosudarstv.
		Functions and limits (the foundations of analysis)", title = "Elementary functions of a real variable.
		The development has the additional benefit of sometimes providing an elementary proof that one program is comparable in stability to another.
	www.math.utah.edu /pub/tex/bib/elefunt.html (1240 words)

	Symbolic Methods
		Differentiating is a mechanical process with a half dozen or so general purpose rules.
		An elementary function is one that can be obtained from rational-valued functions by a finite sequence of nested logarithm, exponential, and algebraic numbers or functions.
		Since √-1 is elementary, all of the "usual" trig and inverse trig functions (sin, cos, arctan) fall into this category since they can be re-expressed using exponentials and logarithms of imaginary numbers.
	www.cs.princeton.edu /introcs/92symbolic (2573 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		I think closed form solution means an elementary function solution.
		An elementary function is a "writable" function, writable only with known terms.
		Each elementary function is described by (complex) constants, polynomials, log and exp and their repeated compositions.
	www.math.niu.edu /~rusin/known-math/99/formal_solut (130 words)

	exam0.pl (Site not responding. Last check: 2007-11-04)
		Word Problem [1] (Elementary Algebra) William has an amount of money in quarters and dimes.
		Algebra Simplify the expression (2-i)/(2+i) into the form Re + i Im (30 Jan 1998).
		Algebra Question, in Frames, with Help If the function f is defined as...
	www.sp.uconn.edu /~cdavid/cgi-bin/exam0.pl?owner=&name= (193 words)

	grad_courses (Site not responding. Last check: 2007-11-04)
		Introduction to complex variables and residue calculus, asymptotic methods, and conformal mapping; integral transforms; ordinary and partial differential equations; non-linear equations; integral equations.
		Elementary kinetic theory: mean free path approach, Boltzmann equation.
		Functional analysis: Banach and HIlbert spaces, linear functionals, operators, and spectral theory.
	physics.bu.edu /grad_courses.html (1221 words)

	M 390 C Differential Algebra (Site not responding. Last check: 2007-11-04)
		This is a theorem of Liouville and we will see a proof of this theorem halfway through this course.
		In general, linear ordinary differential equations (such as y"+2xy'=0) have a "differential Galois group" and the solvability of this group, in the usual group-theoretic sense, is equivalent to the "solvability" of the equation in terms of nice functions in a sense that will be made precise in the course.
		The course will consist of the development of this Differential Galois Theory with some applications.
	www.ma.utexas.edu /users/voloch/diff.html (162 words)

	Electives (Site not responding. Last check: 2007-11-04)
		This course includes functions and graphs; limits and continuity; derivatives of algebraic and trigonometric functions, chain rule; applications to maxima and minima and to related rates; integration; and applications of integration.
		This course includes exponential, logarithmic, circular, and hyperbolic functions: their inverses, derivatives, and integrals; further techniques of integration; improper integrals; limits, L'Hospital's rule; sequences and series, numerical methods; polar coordinates; and introductory differential equations.
		This course covers first-order differential equations, linear differential equations with constant coefficients, first-order systems of differential equations with constant coefficients, numerical methods, Laplace transforms, series solutions; as well as algebraic systems of equations, matrices, determinants, vector spaces, eigenvalues, and eigenvectors.
	www.cs.fit.edu /cs2/catalogs/catalog98/node10.html (1609 words)

	Amazon.ca: Books: Elementary Differential Geometry (Site not responding. Last check: 2007-11-04)
		Although this edition extensively modifies the first edition, it maintains the elementary character of that volume, while providing an introduction to the use of computers and expanding discussion on certain topics.
		My first encounter with this book was during the academic year of 2000-2001, when it was used as the main text for an upper division course on differnetial geometry, at one of the University of California campuses.
		as derivations on the algebra of functions) but this is not appropriate for a first course.
	www.amazon.ca /exec/obidos/ASIN/0125267452 (1511 words)

	[No title]
		Also an elementary function, such as $e^x$ or $sin(x)$, is given by an initial approximating polynomial, together with an algorithm which produces a sequence of increasingly good approximating polynomials, which converge to a limit.
		The elementary numbers form a computable field, and the real elementary numbers form a computable real closed exponential field, unless Schanuel's conjecture is false.
		Alexei Bocharov will present several new algebraic computation features that are implemented in the forthcoming version of Mathematica, including algebraic numbers and algebraic root objects as well as new features in definite integration, enhancements to the Grobner basis function, and substantially improved symbolic and differential equation solving functions.
	ftp.ccs.neu.edu /pub/sigsam/issac95/advance-prog.txt (3642 words)

	Differential Galois theory and algorithms for linear ODE's. (Site not responding. Last check: 2007-11-04)
		The Risch' algorithm involves the study of solutions of the Risch differential equation y'=ay+b over some differential field K, which are either in K or are algebraic over K.
		The eigenring of a differential equation L is the finite dimensional C-algebra of all the endomorphism of the equation.
		This assures that the differential Galois group of the equation is at least a subgroup of G.
	www-lmc.imag.fr /cathode2/Cirm2000/extended/Vanderput/Vanderput.html (2569 words)

	COMPUTER ALGEBRA: LECTURE #12 (Site not responding. Last check: 2007-11-04)
		We shall prove that this integration can be performed by calculations of resultants and gcds and at the same time estimate the least extension field of the rationals that is needed in the calculations.
		Before getting started with actual algorithms, several algebraic notions and results have to be explained: elementary functions, differential fields, elementary function fields etc. Moreover, before integrating, we have to investigate the structure of the derivatives of elementary functions (these are then the functions that possess an elementary integral!)
		the introduction to elementary functions and the Risch algorithm (ch.
	www.math.aau.dk /~raussen/CA/99/lp12/lp12.html (206 words)

	The Math Forum - Math Library - ODE (Site not responding. Last check: 2007-11-04)
		A short article designed to provide an introduction to ordinary differential equations, equations to be solved in which the unknown element is a function rather than a number, and in which the known information relates that function to its derivatives.
		Currently includes presentations and contributed papers from the Fifth Conference on the Teaching of Mathematics held on June 21-22, 1996 in Baltimore, MD and the Sixth Conference on the Teaching of Mathematics held on June 20-21, 1997 in Milwaukee, WI.
		A large collection of images based on or related to mathematical principles, many purely mathematical abstractions, such as fractals; others of natural objects whose shape is explained by equations and mathematical models (for example, a drum vibrating).
	mathforum.org /library/topics/ordinary_diffeq (2302 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		S.R. Nagpaul Graphing Calculator strongly suggested text book, working paper, and pencils are required for each class Prerequisite: Working knowledge of Calculus III and a recent completion of Math 260 with a grade at least 2.0 Course Description: This is a first course in linear algebra and differential equations.
		Techniques of linear algebra are applied to the solution of differential equations.
		Topics coved include first order differential equations and applications, matrices, vector spaces, linear transformations, linear differential equations, systems of differential equations, and Laplace Transforms.
	puma.kvcc.edu /dcunningham/Kvccoutline264Fall2004.doc (532 words)

	The Math Forum - Math Library - Differential Eqtns (Site not responding. Last check: 2007-11-04)
		Elementary bifurcation theory is topic is rarely included in traditional differential equations courses, yet it is of crucial importance in many engineering applications.
		BIT emphasizes numerical methods in approximation, linear algebra, and ordinary and partial differential equations, but also publishes papers in areas such as numerical functional analysis and numerical optimization.
		A computer-based course about calculus, differential equations, and matrix theory, which the instructor can use as soon as the computers are unloaded at the classroom door.
	mathforum.org /library/topics/diffeq (2294 words)

	math lessons - Differential Galois theory (Site not responding. Last check: 2007-11-04)
		The most often encountered example of such a function is exp(-x
		This has the form of a logarithmic derivative.
		algebra arithmetic calculus equations geometry differential equations trigonometry number theory probability theory applied mathematics mathematical games mathematicians
	www.mathdaily.com /lessons/Differential_Galois_theory (548 words)

	Integration and Differential Equations in Computer Algebra - Bronstein (ResearchIndex)
		After a brief review of the mathematical history of those problems, we outline the two major algorithms for them (respectively the Risch and Singer algorithms) and the recent improvements on those algorithms which has allowed them to be implemented.
		1 Introduction An elementary function of a variable x is a...
		2 The Transcendental Risch Differential Equation (context) - Bronstein - 1990 ACM
	citeseer.ist.psu.edu /99367.html (423 words)

	Amazon.com: Elementary Differential Geometry, Second Edition: Books: Barrett O'Neill (Site not responding. Last check: 2007-11-04)
		Modern Differential Geometry for Physicists (World Scientific Lecture Notes in Physics) by C.
		Finally, in the (new to the second edition) chapter eight --which is of a highly topological nature-- the concepts of complete surfaces, covering spaces, Jacobi fields & conjugate points, homogeneous surfaces, and isometric immersions are discussed, followed by two sections exploring the classification of surfaces of constant curvature and the theorems of Bonnet and Hadamard.
		The corresponding concepts of function, differentiability and tangent vectors on these objects is introduced.
	www.amazon.com /exec/obidos/tg/detail/-/0125267452?v=glance (2829 words)

Try your search on: Qwika (all wikis)