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Parabolic Cylinder Function
In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindrical coordinates. The above equation may be brought into two distinct forms (A) and (B) by completing the square and rescaling , called H. F. Weber's equations: and If f(a,z) is a solution, then so are f(a,-z), f(-a,iz)\textf(-a,-iz). If f(a,z)\, is a solution of equation (), then f(-ia,ze^) is a solution of (), and, by symmetry, f(-ia,-ze^), f(ia,-ze^)\textf(ia,ze^) are also solutions of (). Solutions There are independent even and odd solutions of the form (). These are given by (following the notation of Abramowitz and Stegun (1965)): y_1(a;z) = \exp(-z^2/4) \;_1F_1 \left(\tfrac12a+\tfrac14; \; \tfrac12\; ; \; \frac\right)\,\,\,\,\,\, (\mathrm) and y_2(a;z) = z\exp(-z^2/4) \;_1F_1 \left(\tfrac12a+\tfrac34; \; \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabolic Cylindrical Coordinates
In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinates, orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinates, parabolic coordinate system in the perpendicular z-direction. Hence, the Coordinate system#Coordinate surface, coordinate surfaces are confocal parabola, parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of edges. Basic definition The parabolic cylindrical coordinates are defined in terms of the Cartesian coordinates by: :\begin x &= \sigma \tau \\ y &= \frac \left( \tau^2 - \sigma^2 \right) \\ z &= z \end The surfaces of constant form confocal parabolic cylinders : 2 y = \frac - \sigma^2 that open towards , whereas the surfaces of constant form confocal parabolic cylinders : 2 y = -\frac + \tau^2 that open in the opposite direction, i.e., towards . The foci of all these parabolic cylinders are located along ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-integer
In mathematics, a half-integer is a number of the form n + \tfrac, where n is an integer. For example, 4\tfrac12,\quad 7/2,\quad -\tfrac,\quad 8.5 are all ''half-integers''. The name "half-integer" is perhaps misleading, as each integer n is itself half of the integer 2n. A name such as "integer-plus-half" may be more accurate, but while not literally true, "half integer" is the conventional term. Half-integers occur frequently enough in mathematics and in quantum mechanics that a distinct term is convenient. Note that halving an integer does not always produce a half-integer; this is only true for odd integers. For this reason, half-integers are also sometimes called half-odd-integers. Half-integers are a subset of the dyadic rationals (numbers produced by dividing an integer by a power of two). Notation and algebraic structure The set of all half-integers is often denoted \mathbb Z + \tfrac \quad = \quad \left( \tfrac \mathbb Z \right) \smallsetminus \mathbb Z ~. The integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Harmonic Oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known. One-dimensional harmonic oscillator Hamiltonian and energy eigenstates The Hamiltonian of the particle is: \hat H = \frac + \frac k ^2 = \frac + \frac m \omega^2 ^2 \, , where is the particle's mass, is the force constant, \omega = \sqrt is the angular frequency of the oscillator, \hat is the position operator (given by in the coordinate basis), and \hat is the momentum operator (given by \hat p = -i \hbar \, \partial / \partial x in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Polynomial
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as in connection with Brownian motion; * combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * numerical analysis as Gaussian quadrature; * physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term \beginxu_\end is present); * systems theory in connection with nonlinear operations on Gaussian noise. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center ''c'' is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ... [...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] |
Hermite Polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as in connection with Brownian motion; * combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * numerical analysis as Gaussian quadrature; * physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term \beginxu_\end is present); * systems theory in connection with nonlinear operations on Gaussian noise. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in ... [...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] |
Plot Of The Parabolic Cylinder Function D V(z) With V=5 In The Complex Plane From -2-2i To 2+2i With Colors Created With Mathematica 13
Plot or Plotting may refer to: Art, media and entertainment * Plot (narrative), the connected story elements of a piece of fiction Music * ''The Plot'' (album), a 1976 album by jazz trumpeter Enrico Rava * The Plot (band), a band formed in 2003 Other * ''Plot'' (film), a 1973 French-Italian film * ''Plotting'' (video game), a 1989 Taito puzzle video game, also called Flipull * ''The Plot'' (video game), a platform game released in 1988 for the Amstrad CPC and Sinclair Spectrum * ''Plotting'' (non-fiction), a 1939 book on writing by Jack Woodford * ''The Plot'' (novel), a 2021 mystery by Jean Hanff Korelitz * The Plot (card game), a Patience-type card game * The Plot (film), a 2024 South Korean crime thriller film Graphics * Plot (graphics), a graphical technique for representing a data set * Plot (radar), a graphic display that shows all collated data from a ship's on-board sensors * Plot plan, a type of drawing which shows existing and proposed conditions for a given ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abramowitz And Stegun
''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the National Institute of Standards and Technology (NIST). Its full title is ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables''. A digital successor to the Handbook was released as the " Digital Library of Mathematical Functions" (DLMF) on 11 May 2010, along with a printed version, the '' NIST Handbook of Mathematical Functions'', published by Cambridge University Press. Overview Since it was first published in 1964, the 1046-page ''Handbook'' has been one of the most comprehensive sources of information on special functions, containing definitions, identities, approximations, plots, and tables of values of numerous functions used in virtually all fields of applied mathematics. The notation used in the ''Handbook'' is the '' de facto'' standard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completing The Square
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By subsequently isolating and taking the square root, a quadratic problem can be reduced to a linear problem. The name ''completing the square'' comes from a geometrical picture in which represents an unknown length. Then the quantity represents the area of a square of side and the quantity represents the area of a pair of Congruence (geometry), congruent rectangles with sides and . To this square and pair of rectangles one more square is added, of side length . This crucial step ''completes'' a larger square of side length . Completing the square is the oldest method of solving general quadratic equations, used in Old Babylonian Empire, Old Babylonian clay tablets dating from 1800–1600 BCE, and is still taught in elementary algebra c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
