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85 relations: Abramowitz and Stegun, Airy function, Analytic function, Appell sequence, Bell polynomials, Binomial type, C (programming language), Cauchy's integral formula, Charles Hermite, Christoffel–Darboux formula, Combinatorics, Complex number, Confluent hypergeometric function, Contour integration, Correspondence principle, Dirac delta function, Discrete Mathematics (journal), Distribution (mathematics), Double factorial, Edgeworth series, Eigenfunction, Eigenvalues and eigenvectors, Entire function, Expected value, Exponential function, Floor and ceiling functions, Fourier transform, Fractional Fourier transform, Gaussian function, Gaussian noise, Gaussian quadrature, Generating function, GNU Scientific Library, Group (mathematics), Harald Cramér, Hermite number, Hermite transform, Hilbert space, Hilbrand J. Groenewold, Kronecker delta, Laguerre polynomials, Lebesgue measure, Legendre polynomials, Linear span, Lp space, Mathematics, Measure (mathematics), Mehler kernel, Multiplication theorem, Normal distribution, ..., Numerical analysis, Orthogonal polynomials, Orthonormal basis, Orthonormality, Pafnuty Chebyshev, Parabolic cylinder function, Parametric family, Physics, Pierre-Simon Laplace, Polynomial sequence, Power series, Probability, Probability density function, Quantum harmonic oscillator, Quantum state, Random matrix, Random variable, Recurrence relation, Recursion, Right half-plane, Romanovski polynomials, Schrödinger equation, Sheffer sequence, Standard deviation, Stirling's approximation, Systems theory, Taylor series, Telephone number (mathematics), Turán's inequalities, Umbral calculus, Wave function, Weierstrass transform, Weight function, Whittaker function, Wigner distribution function. Expand index (35 more) »
Abramowitz and Stegun
Abramowitz and Stegun (AS) is the informal name of a 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).
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Airy function
In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–92).
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Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series.
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Appell sequence
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \_ satisfying the identity and in which p_0(x) is a non-zero constant.
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Bell polynomials
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions.
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Binomial type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities Many such sequences exist.
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C (programming language)
C (as in the letter ''c'') is a general-purpose, imperative computer programming language, supporting structured programming, lexical variable scope and recursion, while a static type system prevents many unintended operations.
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Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.
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Charles Hermite
Prof Charles Hermite FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
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Christoffel–Darboux formula
In mathematics, the Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by and.
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Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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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.
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Contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
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Correspondence principle
In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers.
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Dirac delta function
In mathematics, the Dirac delta function (function) is a generalized function or distribution introduced by the physicist Paul Dirac.
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Discrete Mathematics (journal)
Discrete Mathematics is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications.
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Distribution (mathematics)
Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.
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Double factorial
In mathematics, the double factorial or semifactorial of a number (denoted by) is the product of all the integers from 1 up to that have the same parity (odd or even) as.
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Edgeworth series
The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants.
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Eigenfunction
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.
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Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
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Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.
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Expected value
In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents.
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Exponential function
In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.
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Floor and ceiling functions
In mathematics and computer science, the floor function is the function that takes as input a real number x and gives as output the greatest integer less than or equal to x, denoted \operatorname(x).
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Fourier transform
The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.
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Fractional Fourier transform
In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform.
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Gaussian function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form: for arbitrary real constants, and.
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Gaussian noise
Gaussian noise is statistical noise having a probability density function (PDF) equal to that of the normal distribution, which is also known as the Gaussian distribution.
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Gaussian quadrature
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration.
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Generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a power series.
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GNU Scientific Library
The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Harald Cramér
Harald Cramér (25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory.
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Hermite number
In mathematics, Hermite numbers are values of Hermite polynomials at zero argument.
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Hermite transform
In mathematics, Hermite transform is an integral transform named after the mathematician Charles Hermite, which uses Hermite polynomials H_n(x) as kernels of the transform.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Hilbrand J. Groenewold
Hilbrand Johannes "Hip" Groenewold (1910–1996) was a Dutch theoretical physicist who pioneered the largely operator-free formulation of quantum mechanics in phase space known as phase-space quantization.
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Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers.
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Laguerre polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are solutions of Laguerre's equation: which is a second-order linear differential equation.
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Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.
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Legendre polynomials
In mathematics, Legendre functions are solutions to Legendre's differential equation: They are named after Adrien-Marie Legendre.
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Linear span
In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space is the intersection of all subspaces containing that set.
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Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.
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Mehler kernel
defined a function and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials (.) based on weight function exp(−²) as This result is useful, in modified form, in quantum physics, probability theory, and harmonic analysis.
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Multiplication theorem
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function.
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Normal distribution
In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution.
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Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
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Orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
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Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.
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Orthonormality
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors.
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Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev (p) (–) was a Russian mathematician.
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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.
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Parametric family
In mathematics and its applications, a parametric family or a parameterized family is a family of objects (a set of related objects) whose differences depend only on the chosen values for a set of parameters.
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Physics
Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.
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Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French scholar whose work was important to the development of mathematics, statistics, physics and astronomy.
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Polynomial sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3,..., in which each index is equal to the degree of the corresponding polynomial.
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Power series
In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.
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Probability
Probability is the measure of the likelihood that an event will occur.
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Probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function, whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.
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Quantum harmonic oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator.
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Quantum state
In quantum physics, quantum state refers to the state of an isolated quantum system.
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Random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables.
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Random variable
In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.
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Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.
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Recursion
Recursion occurs when a thing is defined in terms of itself or of its type.
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Right half-plane
In complex variables, the right half plane is the set of all points in the complex plane whose real part is positive, Category:Complex analysis.
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Romanovski polynomials
In mathematics, Romanovski polynomials is an informal term for one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski in French transcription) within the context of probability distribution functions in statistics.
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Schrödinger equation
In quantum mechanics, the Schrödinger equation is a mathematical equation that describes the changes over time of a physical system in which quantum effects, such as wave–particle duality, are significant.
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Sheffer sequence
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics.
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Standard deviation
In statistics, the standard deviation (SD, also represented by the Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.
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Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials.
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Systems theory
Systems theory is the interdisciplinary study of systems.
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Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
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Telephone number (mathematics)
In mathematics, the telephone numbers or the involution numbers are a sequence of integers that count the ways telephone lines can be connected to each other, where each line can be connected to at most one other line.
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Turán's inequalities
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by (and first published by). There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors.
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Umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to 'prove' them.
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Wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system.
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Weierstrass transform
In mathematics, the Weierstrass transform of a function, named after Karl Weierstrass, is a "smoothed" version of obtained by averaging the values of, weighted with a Gaussian centered at x.
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Weight function
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set.
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Whittaker function
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric.
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Wigner distribution function
The Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis.
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References
[1] https://en.wikipedia.org/wiki/Hermite_polynomials
