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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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and was pursued by Andrey Markov, A. A. Markov and Thomas Joannes Stieltjes, T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (Gaussian quadrature, quadrature rules), probability theory, representation theory (of Lie group, Lie groups, quantum group, quantum groups, and re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called 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 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 varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Filter
In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat Group delay and phase delay, group/phase delay (maximally linear phase response), which preserves the wave shape of filtered signals in the passband. Bessel filters are often used in audio crossover systems. The filter's name is a reference to German mathematician Friedrich Bessel (1784–1846), who developed the mathematical theory on which the filter is based. The filters are also called Bessel–Thomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design in 1949. The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases. While the time-domain step response of the Gaussian filter has zero overshoot (signal), overshoot, the Bessel filter has a small amount of overshoot, but still much less than other common frequency-domain filters, such as Butterworth filters. I ... [...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] |
Confluent Hypergeometric Function
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] |
Laguerre Polynomial
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions only if is a non-negative integer. Sometimes the name Laguerre polynomials is used for solutions of xy'' + (\alpha + 1 - x)y' + ny = 0~. where is still a non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor Nikolay Yakovlevich Sonin). More generally, a Laguerre function is a solution when is not necessarily a non-negative integer. The Laguerre polynomials are also used for Gaussian quadrature to numerically compute integrals of the form \int_0^\infty f(x) e^ \, dx. These polynomials, usually denoted , , …, are a polynomial sequence which may be defined by the Rodrigues formula, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pochhammer Symbol
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \end The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as :\begin x^ = x^\overline &= \overbrace^ \\ &= \prod_^n(x+k-1) = \prod_^(x+k) \,. \end The value of each is taken to be 1 (an empty product) when . These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation , where is a non-negative integer. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used with yet another meaning, namely to d ... [...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 positive i ... [...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, |
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] |
Hankel Transform
In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor ''k'' along the ''r'' axis. The necessary coefficient of each Bessel function in the sum, as a function of the scaling factor ''k'' constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval. Definition The Hankel transform of order \nu of a function ''f''(''r'') is given by : F_\nu(k) = \int_0^\infty f(r) J_\nu(kr) \,r\,\mathrmr, where J_\nu is the Bessel function of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
