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Lommel Polynomial
A Lommel polynomial ''R''''m'',ν(''z'') is a polynomial in 1/''z'' giving the recurrence relation :\displaystyle J_(z) = J_\nu(z)R_(z) - J_(z)R_(z) where ''J''ν(''z'') is a Bessel function of the first kind. They are given explicitly by :R_(z) = \sum_^\frac(z/2)^. See also *Lommel function *Neumann polynomial In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case \alpha=0, are a sequence of polynomials in 1/t used to expand functions in term of Bessel functions. The first few polynomials are :O_0^(t)=\frac 1 t, :O_1^(t) ... References * * Polynomials Special functions {{polynomial-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, 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, which represents 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 solving the Helmholtz equation in spherical coordinates. Applications Bessel's equation arises when finding separa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugen Von Lommel
Eugen Cornelius Joseph von Lommel (19 March 1837, Edenkoben – 19 June 1899, Munich) was a German physicist. He is notable for the Lommel polynomial, the Lommel function, the Lommel–Weber function, and the Lommel differential equation. He is also notable as the doctoral advisor of the Nobel Prize winner Johannes Stark. Lommel was born in Edenkoben in the Palatinate, Kingdom of Bavaria. He studied mathematics and physics at the University of Munich between 1854 and 1858. From 1860 to 1865 he is teacher of physics and chemistry at the canton school of Schwyz. From 1865 to 1867 he taught at the high school in Zürich and was simultaneously Privatdozent at the local university as well as at the polytechnic school. From 1867 to 1868, he was appointed professor of physics at the University of Hohenheim. Finally he was appointed to a chair of experimental physics at Erlangen Erlangen (; , ) is a Middle Franconian city in Bavaria, Germany. It is the seat of the administrativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lommel Function
Lommel () is a Municipalities of Belgium, municipality and City status in Belgium, city in the Belgium, Belgian province of Limburg (Belgium), Limburg. Lying in the Campine, Kempen, it has about 34,000 inhabitants and is part of the arrondissement of Maaseik. Besides the residential town, Lommel also has a number of nature reserves, such as the nature reserve De Watering, the Lommel Sahara, and numerous forests and heathlands. Lommel is the third shopping city in Belgian Limburg with a commercial and shopping center ''De Singel''. Importantly, the silver sand that is mined here for the benefit of the glass industry. Some sand mining quarries are transformed into nature reserves and recreational areas, including Lommel Sahara. The city of Lommel is in the watershed of the basins of the Scheldt, and Meuse, and within these basins Nete (river), Nete Dommel respectively. Tamar (Lommel), Tamar, a maternity home for pregnant girls and unmarried mothers and their children run by the Con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neumann Polynomial
In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case \alpha=0, are a sequence of polynomials in 1/t used to expand functions in term of Bessel functions. The first few polynomials are :O_0^(t)=\frac 1 t, :O_1^(t)=2\frac , :O_2^(t)=\frac + 4\frac , :O_3^(t)=2\frac + 8\frac , :O_4^(t)=\frac + 4\frac + 16\frac . A general form for the polynomial is :O_n^(t)= \frac \sum_^ (-1)^\frac \left(\frac 2 t \right)^, and they have the "generating function" :\frac \frac 1 = \sum_O_n^(t) J_(z), where ''J'' are Bessel functions. To expand a function ''f'' in the form :f(z)=\left(\frac\right)^\alpha \sum_ a_n J_(z)\, for , t, |
Polynomials
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
