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Topic: Hermite polynomials

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	hermite polynomials (Site not responding. Last check: 2007-10-06)
		In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced "air MEET"), are a polynomial sequence defined either by
		Since polynomial sequences form a group under the operation of umbral composition, one may denote by
		Hermite polynomials arise in the theory of Edgeworth series.
	www.yourencyclopedia.net /hermite_polynomials.html (597 words)

	Learn more about Polynomial in the online encyclopedia. (Site not responding. Last check: 2007-10-06)
		The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Weierstrass approximation theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial.
		Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics.
		Depending on the degree of the polynomial to be considered, simply checking if the polynomial has linear factors can eliminate several cases, and then resorting to checking divisibility of some other irreducible polynomials, however Eisenstein's criterion can be used to more efficiently determine irreducibility.
	www.onlineencyclopedia.org /p/po/polynomial.html (1534 words)

	Hermite polynomials (Site not responding. Last check: 2007-10-06)
		History of Polynomial Equations History of quartic, cubic, quantic and quintic polynomials and their solutions.
		Polynomials for all transitive groups up to degree 15, for most of the possible combinations of signature and Galois group.
		Hermite, Charles Mathematician, born at Dieuze, Lorraine, 24 December, 1822; d.
	www.serebella.com /encyclopedia/article-Hermite_polynomials.html (818 words)

	Hermite polynomials - Wikipedia, the free encyclopedia
		In mathematics, the Hermite polynomials, named in honor of Charles Hermite (Hermite is pronounced "air MEET"), are a polynomial sequence defined either by
		The Hermite polynomials constitute an Appell sequence, i.e., they are a polynomial sequence satisfying the identity
		The Hermite polynomials can be expressed as a special case of the Laguerre polynomials.
	en.wikipedia.org /wiki/Hermite_polynomials (638 words)

	Charles Hermite - Wikipedia, the free encyclopedia
		Charles Hermite (pronounced "air meet") (December 24, 1822 - January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
		Hermite polynomials, Hermitian operators, and cubic Hermite splines are named in his honor.
		He was the first to prove that e, the base of natural logarithms, is a transcendental number.
	www.encyclopedia-online.info /Charles_Hermite (116 words)

	Polynomial
		The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial.
		Polynomials are also frequently used to interpolate functions.
		Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones.
	www.brainyencyclopedia.com /encyclopedia/p/po/polynomial.html (1564 words)

	Hermite
		Hermite may have still been an undergraduate but it is likely that his ideas from around 1843 helped Liouville to his important 1844 results which include the result now known as Liouville's theorem.
		Hermite resigned his chair at the École Polytechnique in 1876 but continued to hold the chair at the Sorbonne until he retired in 1897.
		Hermite used to observe [that biology] may be a most useful study even for mathematicians, as hidden and eventually fruitful analogies may appear between processes in both kinds of studies.
	www-history.mcs.st-andrews.ac.uk /history/Mathematicians/Hermite.html (1633 words)

	Appendix: Hermite Polynomial Series (Site not responding. Last check: 2007-10-06)
		The Hermite series is often used to implement cubic splines (de Boor 1978).
		Hermite polynomials form one of the families of classical orthogonal polynomials (Szegö 1975).
		This is one of the reasons why the parameters associated with the polynomials are extremely difficult to interpret.
	jsekhon.fas.harvard.edu /nairu/node2.html (87 words)

	Charles Hermite - Wikipedia, the free encyclopedia
		Charles Hermite (pronounced in IPA, /ˌʃaʁl ɛʁˈmit/, or phonetically "air-meet") (December 24, 1822 - January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
		Hermite polynomials, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor.
		Upon Weierstrass' discovery in 1861 of continuous curves that are nowhere differentiable - they possess no tangent at any point - Hermite famously remarked: “I turn aside with a shudder of horror from this lamentable plague of functions which have no derivatives.”
	www.wikipedia.org /wiki/Hermite (178 words)

	Talk:Hermite polynomials - Wikipedia, the free encyclopedia
		This article needs to have some discussion of the relationship of Hermite polynomials to the eigenfunctions of Schrodinger's equation for the quantum harmonic oscillator, since that is one of the major problems in which they arise.
		Well, I thought Charles Hermite himself introduced the Hermite polynomials, and he died in 1901, when quantum mechanics was just getting started.
		The solutions to the harmonic oscillator do use Hermite polynomials, but perhaps a mentioning in passing is sufficient.
	en.wikipedia.org /wiki/Talk:Hermite_polynomials (991 words)

	Hermite polynomials (Site not responding. Last check: 2007-10-06)
		In mathematics, the Hermite polynomials, named in honorof Charles Hermite (pronounced "air MEET"), are a polynomial sequence defined either by
		These are Hermite polynomial sequences of different variances; see the material on variancesbelow.
		Since polynomial sequences form a group under theoperation of umbral composition, one may denote by
	www.therfcc.org /hermite-polynomials-210622.html (530 words)

	Publications -- Mourad E.H. Ismail
		Polynomials orthogonal with respect to discrete convolution, joint with W. Al-Salam, J. Math.
		On sieved orthogonal polynomials V: Sieved Pollaczek polynomials, joint with J. Charris, SIAM J. Math.
		On sieved orthogonal polynomials VI: Differential equations, joint with J. Bustoz and J. Wimp, Differential and Integral Equations 3 (1990), 757-766.
	www.math.usf.edu /~ismail/publications.html (3529 words)

	Abstract of: The role of Hermite polynomials in asymptotic analysis (Site not responding. Last check: 2007-10-06)
		Hermite polynomials are considered as approximants in asymptotic representations of certain other polynomials.
		Examples are given for polynomials from the Askey scheme of hypergeometric orthogonal polynomials.
		We also mention that Hermite polynomials can be used as main approximants in uniform asymptotic representations of certain types of integrals and differential equations.
	db.cwi.nl /rapporten/abstract.php?abstractnr=825 (84 words)

	PlanetMath: Hermite equation
		Therefore the series converge in the whole complex plane and define entire functions.
		The polynomial solutions of the Hermite equation are usually normed so that the highest degree term is
		This is version 11 of Hermite equation, born on 2005-05-15, modified 2005-10-03.
	www.planetmath.org /encyclopedia/HermiteEquation.html (108 words)

	Hermite Curve Interpolation (Site not responding. Last check: 2007-10-06)
		Hermite curves are very easy to calculate but also very powerful.
		We'll lose some of the flexibility of the hermite curves, but as a tradeoff the curves will be much easier to use.
		They share one thing with the hermite curves: They are still cubic polynomials, but the way they are calculated is different.
	www.cubic.org /~submissive/sourcerer/hermite.htm (1073 words)

	Quantum Harmonic Oscillator (Site not responding. Last check: 2007-10-06)
		But as the quantum number increases, the probabability distribution becomes more like that of the classical oscillator - this tendency to approach the classical behavior for high quantum numbers is called the correspondence principle.
		When the Schrodinger equation for the harmonic oscillator is solved by a series method, the solutions contain this set of polynomials, named the Hermite polynomials.
		In the wavefunction associated with a given value of the quantum number n, the Gaussian is multiplied by a polynomial of order n (the Hermite polynomials above) and the constants necessary to normalize the wavefunctions.
	hyperphysics.phy-astr.gsu.edu /hbase/quantum/hosc5.html (254 words)

	4Reference \|\| Hermite polynomials (Site not responding. Last check: 2007-10-06)
		Definition In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced "air MEET"), compose a polynomial sequence defined either by
		Orthogonality The nth function in this list is an nth-degree polynomial for n = 0, 1, 2, 3,....
		Generalization The Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution
	www.4reference.net /encyclopedias/wikipedia/Hermite_polynomials.html (471 words)

	Series Solutions: Hermite's Equation (Site not responding. Last check: 2007-10-06)
		We know from the previous section that this equation will have series solutions which both converge and solve the differential equation everywhere.
		Hermite's Equation is our first example of a differential equation, which has a polynomial solution.
		Consequently, the solution has only 3 non-zero coefficients, and hence is a polynomial.
	www.sosmath.com /diffeq/series/series06/series06.html (356 words)

	Laguerre polynomials (Site not responding. Last check: 2007-10-06)
		The sequence of Laguerre polynomials is a Sheffer sequence.
		The generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator, due to their Relation to the Hermite polynomials, which can be expressed as :
		The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as :
	laguerre-polynomials.area51.ipupdater.com (291 words)

	Abstract of: Approximations of orthogonal polynomials in terms of Hermite polynomials (Site not responding. Last check: 2007-10-06)
		Abstract of: Approximations of orthogonal polynomials in terms of Hermite polynomials
		Several orthogonal polynomials have limit forms in which Hermite polynomials show up.
		In this paper we are interested in more details of these limits and we give asymptotic representations of several orthogonal polynomials in terms of Hermite polynomials.
	db.cwi.nl /rapporten/abstract.php?abstractnr=800 (155 words)

	Hermite polynomials (Site not responding. Last check: 2007-10-06)
		In mathematics, the Hermite polynomials, named in honor of Charles Hermite (pronounced "air MEET"), are a polynomial sequence defined either by :
		These polynomials are orthogonal with respect to the measure :
		The nth Hermite polynomial satisfies Hermite's differential equation: :
	hermite-polynomials.area51.ipupdater.com (628 words)

	[No title] (Site not responding. Last check: 2007-10-06)
		and a polynomial of degree n and parity (-1)
		The parity of the Hermite polynomials is (-1)
		(z) is a polynomial of degree n-1, the H
	electron6.phys.utk.edu /qm1/modules/m8/eigenfunction.htm (195 words)

	Hermite
		Charles Hermite's father was Ferdinand Hermite and his mother was Madeleine Lallemand.
		On 14 July 1856 Hermite was elected to the
		His working hours were devoted to mathematical research and teaching.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Hermite.html (1633 words)

	Orthogonal Polynomials
		Section 3.2.10 discusses the generalization of Legendre polynomials to Legendre functions, which can have non-integer degrees.
		Series of Chebyshev polynomials are often used in making numerical approximations to functions.
		Legendre, Gegenbauer and Chebyshev polynomials can all be viewed as special cases of Jacobi polynomials.
	documents.wolfram.com /v4/MainBook/3.2.9.html (240 words)

	Lagrange shape Functions _____ (Site not responding. Last check: 2007-10-06)
		The first three are Lagrange functions and the last shows Hermite cubic polynomials.
		The shape functions are used to find the field variable U from known values at other locations.
		The Hermite shape functions are used for beam analysis where both the deflection and slope of adjacent elements must be the same at each node.
	www.nmt.edu /~es421/ansys/shapefnt.htm (164 words)

	Hermite Polynomials
		Here are the first seven Hermite Polynomials, along with a plot.
		The eigenvalues of Hermite's equation are the nonnegative integers 2n.
		Again, by one of the major Sturm-Liouville theorems, the Hermite polynomials are orthogonal with respect to the weight Exp[-x^2].
	www.ma.iup.edu /projects/CalcDEMma/sturm/sl8.html (258 words)

	Citations: the theory and computation of non-perfect Pad'e-Hermite approximants - Cabay, Labahn, Beckermann ... (Site not responding. Last check: 2007-10-06)
		....at infinity satisfying S(1) I, which includes the monic systems of polynomials introduced by Mahler [29] for the case of scalar Hermite Pad e and simultaneous Pad e approximants and [25] for the matrix case.
		Note that the Matrix Simultaneous Pad e Form is closely connected to a rectangular Matrix Hermite Pad e Form if the interpolation conditions are written as follows: 2 6 6 6 6 6 6 6 4 A T 1 (z) A T 2....
		....Pad e approximants were first used by Hermite in 1873 in his famous proof of the transcendence of e.
	citeseer.ist.psu.edu /context/14659/0 (1824 words)

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