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Lommel Polynomial
A Lommel polynomial ''R''''m'',ν(''z''), introduced by , 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 * * *{{citation, first=Eugen von , last=Lommel, title=Zur Theorie der Bessel'schen Functionen , journal =Mathematische Annalen , publisher =Springer , place=Berlin / Heidelberg , volume =4, issue= 1 , year= 1871 , doi =10.1007/BF01443302 , pages =103–116 Polynomials Special functions ... [...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, 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] |
Lommel Function
The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation: : z^2 \frac + z \frac + (z^2 - \nu^2)y = z^. Solutions are given by the Lommel functions ''s''μ,ν(''z'') and ''S''μ,ν(''z''), introduced by , :s_(z) = \frac \left Y_ (z) \! \int_^ \!\! x^ J_(x) \, dx - J_\nu (z) \! \int_^ \!\! x^ Y_(x) \, dx \right :S_(z) = s_(z) + 2^ \Gamma\left(\frac\right) \Gamma\left(\frac\right) \left(\sin \left \mu - \nu)\frac\rightJ_\nu(z) - \cos \left \mu - \nu)\frac\rightY_\nu(z)\right), where ''J''ν(''z'') is a Bessel function of the first kind and ''Y''ν(''z'') a Bessel function of the second kind. See also * Anger function * Lommel polynomial A Lommel polynomial ''R'm'',ν(''z''), introduced by , 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 ... * Struve function * Weber f ... [...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)=\sum_ a_n J_(z)\, for , z, |
Polynomials
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. 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 problem (mathematics education), 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; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic variety ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
