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Multivariate Gamma Function
In mathematics, the multivariate gamma function Γ''p'' is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the probability density function of the Wishart and inverse Wishart distributions, and the matrix variate beta distribution. It has two equivalent definitions. One is given as the following integral over the p \times p positive-definite real matrices: : \Gamma_p(a)= \int_ \exp\left( -(S)\right)\, \left, S\^ dS, where , S, denotes the determinant of S. The other one, more useful to obtain a numerical result is: : \Gamma_p(a)= \pi^\prod_^p \Gamma(a+(1-j)/2). In both definitions, a is a complex number whose real part satisfies \Re(a) > (p-1)/2. Note that \Gamma_1(a) reduces to the ordinary gamma function. The second of the above definitions allows to directly obtain the recursive relationships for p\ge 2: : \Gamma_p(a) = \pi^ \Gamma(a) \Gamma_(a-\tfrac) = \pi^ \Gamma_(a) \Gamma(a+(1-p)/2). Thus * \Gamma_2(a)=\pi^\Gamma(a)\Gam ... [...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] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M \ (z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygamma Function
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: :\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z). Thus :\psi^(z) = \psi(z) = \frac holds where is the digamma function and is the gamma function. They are holomorphic on \mathbb \backslash\mathbb_. At all the nonpositive integers these polygamma functions have a pole of order . The function is sometimes called the trigamma function. Integral representation When and , the polygamma function equals :\begin \psi^(z) &= (-1)^\int_0^\infty \frac\,\mathrmt \\ &= -\int_0^1 \frac(\ln t)^m\,\mathrmt\\ &= (-1)^m!\zeta(m+1,z) \end where \zeta(s,q) is the Hurwitz zeta function. This expresses the polygamma function as the Laplace transform of . It follows from Bernstein's theorem on monotone functions that, for and real and non-negative, is a completely monotone function. Setting in the above formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digamma Function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac. It is the first of the polygamma functions. It is strictly increasing and strictly concave on (0,\infty). The digamma function is often denoted as \psi_0(x), \psi^(x) or (the uppercase form of the archaic Greek consonant digamma meaning double-gamma). Relation to harmonic numbers The gamma function obeys the equation :\Gamma(z+1)=z\Gamma(z). \, Taking the derivative with respect to gives: :\Gamma'(z+1)=z\Gamma'(z)+\Gamma(z) \, Dividing by or the equivalent gives: :\frac=\frac+\frac or: :\psi(z+1)=\psi(z)+\frac Since the harmonic numbers are defined for positive integers as :H_n=\sum_^n \frac 1 k, the digamma function is related to them by :\psi(n)=H_-\gamma, where and is the Euler–Mascheroni constant. For half-integer arguments the digamma function takes the values : \psi \left(n+\tfrac12\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Richards (statistician)
Donald St. P. Richards (born 1955, in Mandeville, Jamaica) is an American statistician conducting research on multivariate statistics, zonal polynomials, distance correlation, total positivity, and hypergeometric functions of matrix argument. He currently serves as a distinguished professor of statistics at the Pennsylvania State University, and is a Fellow of the Institute of Mathematical Statistics and a Fellow of the American Mathematical Society. Richards obtained his PhD in 1978 at the University of the West Indies, where the statistician Rameshwar D. Gupta was his doctoral advisor. In 1999, he was elected a Fellow of the Institute of Mathematical Statistics. In 2012, he was elected a Fellow of the American Mathematical Society. Personal life Richards became an American citizen in 1990. He was married to Mercedes Richards Mercedes Tharam Richards ( Kingston, 14 May 1955 – Hershey, 3 February 2016), née Davis, was a Jamaican astronomy and astrophysics professor. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Library Of Mathematical Functions
The Digital Library of Mathematical Functions (DLMF) is an online project at the National Institute of Standards and Technology (NIST) to develop a database of mathematical reference data for special functions and their applications. It is intended as an update of '' Abramowitz's and Stegun's Handbook of Mathematical Functions'' (A&S). It was published online on 7 May 2010, though some chapters appeared earlier. In the same year it appeared at Cambridge University Press under the title ''NIST Handbook of Mathematical Functions''. In contrast to A&S, whose initial print run was done by the U.S. Government Printing Office and was in the public domain, NIST asserts that it holds copyright to the DLMF under Title 17 USC 105 of the U.S. Code. See also * NIST Dictionary of Algorithms and Data Structures The NIST ''Dictionary of Algorithms and Data Structures'' is a reference work maintained by the U.S. National Institute of Standards and Technology. It defines a large number of te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prasanta Chandra Mahalanobis
Prasanta Chandra Mahalanobis OBE, FNA, FASc, FRS (29 June 1893– 28 June 1972) was an Indian scientist and statistician. He is best remembered for the Mahalanobis distance, a statistical measure, and for being one of the members of the first Planning Commission of free India. He made pioneering studies in anthropometry in India. He founded the Indian Statistical Institute, and contributed to the design of large-scale sample surveys. For his contributions, Mahalanobis has been considered the father of modern statistics in India. Early life Mahalanobis belonged to a family of Bengali landed gentry who lived in Bikrampur (now in Bangladesh). His grandfather Gurucharan (1833–1916) moved to Calcutta in 1854 and built up a business, starting a chemist shop in 1860. Gurucharan was influenced by Debendranath Tagore (1817–1905), father of the Nobel Prize-winning poet, Rabindranath Tagore. Gurucharan was actively involved in social movements such as the Brahmo Samaj, acting as i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wishart (statistician)
John Wishart (28 November 1898 – 14 July 1956) was a Scottish mathematician and agricultural statistician. He gave his name to the Wishart distribution in statistics. Life Wishart was born in Perth, Scotland on 28 November 1898, the son of Elizabeth and John Wishart of Montrose. His father was a bootmaker. The family moved from Montrose to Perth around 1903, living at 36 Robertsons Buildings on Barrack Street. He was educated at Perth Academy. In the First World War he was conscripted into the Black Watch in 1917 and served two years in France. He studied Mathematics at Edinburgh University under Edmund Taylor Whittaker, graduating MA BSc before winning a place at Cambridge University where he gained a further MA. He then gained a doctorate (DSc) at the University College London under Karl Pearson. After a year of teacher training at Moray College of Education in Edinburgh he then worked for some years as a Mathematics Teacher at West Leeds High School. In 1927 he join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indefinite Product
In mathematics, the indefinite product operator is the inverse operator of Q(f(x)) = \frac. It is a discrete version of the geometric integral of geometric calculus, one of the non-Newtonian calculi. Some authors use term discrete multiplicative integration. Thus :Q\left( \prod_x f(x) \right) = f(x) \, . More explicitly, if \prod_x f(x) = F(x) , then :\frac = f(x) \, . If ''F''(''x'') is a solution of this functional equation for a given ''f''(''x''), then so is ''CF''(''x'') for any constant ''C''. Therefore, each indefinite product actually represents a family of functions, differing by a multiplicative constant. Period rule If T is a period of function f(x) then :\prod _x f(Tx)=C f(Tx)^ Connection to indefinite sum Indefinite product can be expressed in terms of indefinite sum: :\prod _x f(x)= \exp \left(\sum _x \ln f(x)\right) Alternative usage Some authors use the phrase "indefinite product" in a slightly different but related way to describe a product in which th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M \ (z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barnes G-function
In mathematics, the Barnes G-function ''G''(''z'') is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be written in terms of the double gamma function. Formally, the Barnes ''G''-function is defined in the following Weierstrass product form: : G(1+z)=(2\pi)^ \exp\left(- \frac \right) \, \prod_^\infty \left\ where \, \gamma is the Euler–Mascheroni constant, exp(''x'') = ''e''''x'' is the exponential function, and Π denotes multiplication (capital pi notation). As an entire function, ''G'' is of order two, and of infinite type. This can be deduced from the asymptotic expansion given below. Functional equation and integer arguments The Barnes ''G''-function satisfies the functional equation : G(z+1)=\Gamma(z)\, G(z) with normalisation ''G''(1) = 1. Note the similarity between the functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive-definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
