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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)\G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined for all complex numbers z except non-positive integers, and for every positive integer z=n, \Gamma(n) = (n-1)!\,.The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: \Gamma(z) = \int_0^\infty t^ e^\textt, \ \qquad \Re(z) > 0\,.The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic function, holomorphic except at zero and the negative integers, where it has simple Zeros and poles, poles. The gamma function has no zeros, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential functi ... [...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 ... [...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(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing and Concave function, strictly concave on (0,\infty), and it Asymptotic analysis, asymptotically behaves as :\psi(z) \sim \ln - \frac, for complex numbers with large modulus (, z, \rightarrow\infty) in the Circular sector, sector , \arg z, 0. The digamma function is often denoted as \psi_0(x), \psi^(x) or (the uppercase form of the archaic Greek consonant digamma meaning Gamma, double-gamma). Gamma. Relation to harmonic numbers The gamma function obeys the equation :\Gamma(z+1)=z\Gamma(z). \, Taking the logarithm on both sides and using the functional equation property of the log-gamma function gives: :\log \Gamma(z+1)=\log(z)+\log \Gamma(z), Differentiating both sides with respect to gives: :\psi(z+1)=\psi(z)+\frac Since the ... [...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 The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Library Of Mathematical Functions
Digital usually refers to something using discrete digits, often binary digits. Businesses *Digital bank, a form of financial institution *Digital Equipment Corporation (DEC) or Digital, a computer company *Digital Research (DR or DRI), a software company Computing and technology Hardware *Digital electronics, electronic circuits which operate using digital signals **Digital camera, which captures and stores digital images *** Digital versus film photography **Digital computer, a computer that handles information represented by discrete values **Digital recording, information recorded using a digital signal Socioeconomic phenomena *Digital culture, the anthropological dimension of the digital social changes *Digital divide, a form of economic and social inequality in access to or use of information and communication technologies * Digital economy, an economy based on computing and telecommunications resources *Digital rights, legal rights of access to computers or the Internet O ... [...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 statistics in India. Since 2007, every year June 29 is celebrated as National Statistics Day in India to commemorate the birth anniversary of P.C. Mahalanobis and his contributions to statistical science and planning. Early life Mahalanobis was born on 29 June 1893, in Calcutta, Bengal Presidency (now West Bengal). Mahalanobis belonged to a prominent Bengali Brahmin family of landed gentry in Bikrampur, Dhaka, Bengal Presidency (now in Bangladesh). His grandfather Guruchar ... [...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 Scott 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 the University of Edinburgh under Edmund Taylor Whittaker, graduating with an MA and BSc. He then went on to study at the University of Cambridge 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 ... [...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. 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 the numerical value of the upper limit is not given. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined for all complex numbers z except non-positive integers, and for every positive integer z=n, \Gamma(n) = (n-1)!\,.The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: \Gamma(z) = \int_0^\infty t^ e^\textt, \ \qquad \Re(z) > 0\,.The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic function, holomorphic except at zero and the negative integers, where it has simple Zeros and poles, poles. The gamma function has no zeros, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barnes G-function
In mathematics, the Barnes G-function ''G''(''z'') is a function (mathematics), 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, exponential function, exp(''x'') = ''e''''x'' is the exponential function, and Π denotes multiplication (capital pi notation). The integral representation, which may be deduced from the relation to the double gamma function, is : \log G(1+z) = \frac\log(2\pi) +\int_0^\infty\frac\left[\frac +\frace^ -\frac\right] As an entire function, ''G'' is of order two, and of infinite type. This can be deduced from the asymptoti ... [...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 \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number \mathbf^* M \mathbf is positive for every nonzero complex column vector \mathbf, where \mathbf^* denotes the conjugate transpose of \mathbf. Positive semi-definite matrices are defined similarly, except that the scalars \mathbf^\mathsf M \mathbf and \mathbf^* M \mathbf 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''. Some authors use more general definitions of definiteness, permitting the matrices to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
