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Padé Table
In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximants :''R''''m'', ''n'' to a given complex formal power series. Certain sequences of approximants lying within a Padé table can often be shown to correspond with successive convergents of a continued fraction representation of a holomorphic or meromorphic function. History Although earlier mathematicians had obtained sporadic results involving sequences of rational approximations to transcendental functions, Frobenius (in 1881) was apparently the first to organize the approximants in the form of a table. Henri Padé further expanded this notion in his doctoral thesis ''Sur la representation approchee d'une fonction par des fractions rationelles'', in 1892. Over the ensuing 16 years Padé published 28 additional papers exploring the properties of his table, and relating the table to analytic continued fractions. Modern interest in Padé tables was revived by H. S. Wall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Padé
Henri Eugène Padé (; 17 December 1863 – 9 July 1953) was a French mathematician, who is now remembered mainly for his development of Padé approximation techniques for functions using rational functions. Education and career Padé studied at École Normale Supérieure in Paris. He then spent a year at Leipzig University and University of Göttingen, where he studied under Felix Klein and Hermann Schwarz. In 1890 Padé returned to France, where he taught in Lille while preparing his doctorate under Charles Hermite. His doctoral thesis described what is now known as the Padé approximant. He then became an assistant professor at Université Lille Nord de France, where he succeeded Émile Borel as a professor of rational mechanics at École centrale de Lille. Padé taught at Lille until 1902, when he moved to Université de Poitiers The University of Poitiers (UP; french: Université de Poitiers) is a public university located in Poitiers, France. It is a membe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple :(x,y,z)=(1,-2,-2), since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chelsea Publishing Company
The Chelsea Publishing Company was a publisher of mathematical books, based in New York City, founded in 1944 by Aaron Galuten while he was still a graduate student at Columbia Columbia may refer to: * Columbia (personification), the historical female national personification of the United States, and a poetic name for America Places North America Natural features * Columbia Plateau, a geologic and geographic region i .... Its initial focus was to republish important European works that were unavailable in the United States because of wartime restrictions, such as Hausdorff's Mengenlehre, or because the works were out of print. This soon expanded to include translations of such works into English, as well as original works by American authors. As of 1985, the company's catalog included more than 200 titles. After Galuten's death in 1994, the company was acquired in 1997 by the AMS, which continues to publish a portion of the company's original catalog under the ''AMS Chelsea' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shanks Transformation
In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941.Weniger (2003). Formulation For a sequence \left\_ the series :A = \sum_^\infty a_m\, is to be determined. First, the partial sum A_n is defined as: :A_n = \sum_^n a_m\, and forms a new sequence \left\_. Provided the series converges, A_n will also approach the limit A as n\to\infty. The Shanks transformation S(A_n) of the sequence A_n is the new sequence defined byBender & Orszag (1999), pp. 368–375.Van Dyke (1975), pp. 202–205. :S(A_n) = \frac = A_ - \frac where this sequence S(A_n) often converges more rapidly than the sequence A_n. Further speed-up may be obtained by repeated use of the Shanks transformation, by computing S^2(A_n)=S(S(A_n)), S^3(A_n)=S(S(S(A_n))), etc. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Newton Series
Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements (forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal attire, attire for semi-formal events * Informal attire, more controlled attire than casual but less than formal * Formal (university), official university dinner, ball or other event * School formal, official school dinner, ball or other event Logic and mathematics *Formal logic, or mathematical logic ** Informal logic, the complement, whose definition and scope is contentious *Formal fallacy, reasoning of invalid structure ** Informal fallacy, the complement *Informal mathematics, also called naïve mathematics *Formal cause, Aristotle's intrinsic, determining cause * Formal power series, a generalization of power series without requiring convergence, used in combinatorics *Formal calculation, a calculation which is systematic, but without a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Recurrence Formulas
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values. A generalized continued fraction is an expression of the form :x = b_0 + \cfrac where the () are the partial numerators, the are the partial denominators, and the leading term is called the ''integer'' part of the continued fraction. The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas: :\begin x_0 &= \frac = b_0, \\ pxx_1 &= \frac = \frac, \\ pxx_2 &= \frac = \frac,\ \dots \end where is the ''numerator'' and is the ''denominator'', called continuants, of the th convergent. They are given by the recursion :\begin A_n &= b_n A_ + a_n A_, \\ B_n &= b_n B_ + a_n B_ \qquad \text n \ge 1 \end with initial values :\begin A_ &= 1,& A_0&=b_0,\\ B_&=0, & B_0&=1. \end If the sequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Confluent Hypergeometric Series
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The term ''confluent'' refers to the merging of singular points of families of differential equations; ''confluere'' is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions: * Kummer's (confluent hypergeometric) function , introduced by , is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated Kummer's function bearing the same name. * Tricomi's (confluent hypergeometric) function introduced by , sometimes denoted by , is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind. * Whittaker functions (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss's Continued Fraction
In complex analysis, Gauss's continued fraction is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several important elementary functions, as well as some of the more complicated transcendental functions. History Lambert published several examples of continued fractions in this form in 1768, and both Euler and Lagrange investigated similar constructions, but it was Carl Friedrich Gauss who utilized the algebra described in the next section to deduce the general form of this continued fraction, in 1813. Although Gauss gave the form of this continued fraction, he did not give a proof of its convergence properties. Bernhard Riemann and L.W. Thomé obtained partial results, but the final word on the region in which this continued fraction converges was not given until 1901, by Edward Burr Van Vleck. Derivation Let f_0, f_1, f_2, \dots be a seque ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Polynomials
In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series :y_n(x)=\sum_^n\frac\,\left(\frac\right)^k. Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials :\theta_n(x)=x^n\,y_n(1/x)=\sum_^n\frac\,\frac. The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is :y_3(x)=15x^3+15x^2+6x+1 while the third-degree reverse Bessel polynomial is :\theta_3(x)=x^3+6x^2+15x+15. The reverse Bessel polynomial is used in the design of Bessel electronic filters. Properties Definition in terms of Bessel functions The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name. :y_n(x)=\,x^\theta_n(1/x)\, :y_n(x)=\sqrt\,e^K_(1/x) :\theta_n(x)=\sqrt\,x^e^K_(x) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Hypergeometric Series
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. Notation A hypergeometric series is formally defined as a power series :\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_ \beta_n z^n in which the ratio of successive coefficients ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics". The exponential function satisfies the exponentiation identity e^ = e^x e^y \text x,y\in\mathbb, which, along with the definition e = \exp(1), shows that e^n=\underbrace_ for positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
