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Hahn–Exton Q-Bessel Function
In mathematics, the Hahn–Exton ''q''-Bessel function or the third Jackson q-Bessel function, Jackson ''q''-Bessel function is a q-analog, ''q''-analog of the Bessel function, and satisfies the Hahn-Exton ''q''-difference equation (). This function was introduced by in a special case and by in general. The Hahn–Exton ''q''-Bessel function is given by : J_\nu^(x;q) = \frac \sum_\frac= \frac x^\nu _1\phi_1(0;q^;q,qx^2). \phi is the basic hypergeometric function. Properties Zeros Koelink and Swarttouw proved that J_\nu^(x;q) has infinite number of real zeros. They also proved that for \nu>-1 all non-zero roots of J_\nu^(x;q) are real (). For more details, see and . Zeros of the Hahn-Exton ''q''-Bessel function appear in a discrete analog of Daniel Bernoulli's problem about free vibrations of a lump loaded chain (, ) Derivatives For the (usual) derivative and ''q''-derivative of J_\nu^(x;q), see . The symmetric ''q''-derivative of J_\nu^(x;q) is described on . Recurrence R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jackson Q-Bessel Function
In mathematics, a Jackson ''q''-Bessel function (or basic Bessel function) is one of the three ''q''-analogs of the Bessel function introduced by . The third Jackson ''q''-Bessel function is the same as the Hahn–Exton ''q''-Bessel function. Definition The three Jackson ''q''-Bessel functions are given in terms of the ''q''-Pochhammer symbol and the basic hypergeometric function \phi by : J_\nu^(x;q) = \frac (x/2)^\nu _2\phi_1(0,0;q^;q,-x^2/4), \quad , x, -1, the second Jackson ''q''-Bessel function satisfies: \left, J_^(z;q)\\leq\frac\left(\frac\right)^\nu\exp\left\. (see .) For n\in\mathbb, \left, J_^(z;q)\\leq\frac\left(\frac\right)^n(-, z, ^2;q)_. (see .) Generating Function The following formulas are the ''q''-analog of the generating function for the Bessel function (see ): :\sum_^t^nJ_n^(x;q)=(-x^2/4;q)_e_q(xt/2)e_q(-x/2t), :\sum_^t^nJ_n^(x;q)=e_q(xt/2)E_q(-qx/2t). e_q is the ''q''-exponential function. Alternative Representations Integral Representations The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-analog
In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q''-analogs that arise naturally, rather than in arbitrarily contriving ''q''-analogs of known results. The earliest ''q''-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, , , ''q''-analogues are most frequently studied in the mathematical fields of combinatorics and special functions. In these settings, the limit is often formal, as is often discrete-valued (for example, it may represent a prime power). ''q''-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of chaotic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basic Hypergeometric Function
In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x''''n'' is called hypergeometric if the ratio of successive terms ''x''''n''+1/''x''''n'' is a rational function of ''n''. If the ratio of successive terms is a rational function of ''q''''n'', then the series is called a basic hypergeometric series. The number ''q'' is called the base. The basic hypergeometric series _2\phi_1(q^,q^;q^;q,x) was first considered by . It becomes the hypergeometric series F(\alpha,\beta;\gamma;x) in the limit when base q =1. Definition There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ. The unilateral basic hypergeometric series is defined as :\;_\phi_k \left begin a_1 & a_2 & \ldots & a_ \\ b_1 & b_2 & \ldots & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Bernoulli
Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. His name is commemorated in the Bernoulli's principle, a particular example of the conservation of energy, which describes the mathematics of the mechanism underlying the operation of two important technologies of the 20th century: the carburetor and the airplane wing. Early life Daniel Bernoulli was born in Groningen, in the Netherlands, into a family of distinguished mathematicians. Rothbard, MurrayDaniel Bernoulli and the Founding of Mathematical Economics ''Mises Institute'' (excerpted from ''An Austrian Perspective on the History of Economic Thought'') The Bernoulli family came originally from Antwerp, at that time in the Spanish Netherlands, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Journal Of Mathematics And Mathematical Sciences
The ''International Journal of Mathematics and Mathematical Sciences'' is a biweekly peer-reviewed mathematics journal. It was established in 1978 by Lokenath Debnath and is published by the Hindawi Publishing Corporation. The journal publishes articles in all areas of mathematics such as pure and applied mathematics, mathematical physics, theoretical mechanics, probability and mathematical statistics, and theoretical biology. Indexing and abstracting The journal is or has been indexed and abstracted in the following bibliographic databases: *EBSCO Information Services *Emerging Sources Citation Index *Mathematical Reviews * ProQuest databases *Scopus *Zentralblatt MATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructur ... References External links *Website prior to 3 March 2001 P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analysis Mathematica
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 B.C.), though ''analysis'' as a formal concept is a relatively recent development. The word comes from the Ancient Greek ἀνάλυσις (''analysis'', "a breaking-up" or "an untying;" from ''ana-'' "up, throughout" and ''lysis'' "a loosening"). From it also comes the word's plural, ''analyses''. As a formal concept, the method has variously been ascribed to Alhazen, René Descartes (''Discourse on the Method''), and Galileo Galilei. It has also been ascribed to Isaac Newton, in the form of a practical method of physical discovery (which he did not name). The converse of analysis is synthesis: putting the pieces back together again in new or different whole. Applications Science The field of chemistry uses analysis in thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Mathematical Analysis And Applications
The ''Journal of Mathematical Analysis and Applications'' is an academic journal in mathematics, specializing in mathematical analysis and related topics in applied mathematics. It was founded in 1960, as part of a series of new journals on areas of mathematics published by Academic Press, and is now published by Elsevier Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as '' The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', .... For most years since 1997 it has been ranked by SCImago Journal Rank as among the top 50% of journals in its topic areas. retrieved 201 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Journal Of Mathematics
The ''Canadian Journal of Mathematics'' (french: Journal canadien de mathématiques) is a bimonthly mathematics journal published by the Canadian Mathematical Society. It was established in 1949 by H. S. M. Coxeter and G. de B. Robinson. The current editors-in-chief of the journal are Louigi Addario-Berry and Eyal Goren. The journal publishes articles in all areas of mathematics. See also * Canadian Mathematical Bulletin The ''Canadian Mathematical Bulletin'' (french: Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antoni ... References External links * University of Toronto Press academic journals Mathematics journals Publications established in 1949 Bimonthly journals Multilingual journals Cambridge University Press academic journals Academic journals associated with learned and professional societies of Canada ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the List of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
