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Double Gamma Function
In mathematics, the multiple gamma function \Gamma_N is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was studied by . At the end of this paper he mentioned the existence of multiple gamma functions generalizing it, and studied these further in . Double gamma functions \Gamma_2 are closely related to the q-gamma function, and triple gamma functions \Gamma_3 are related to the elliptic gamma function. Definition For \Re a_i>0, let :\Gamma_N(w\mid a_1,\ldots,a_N) = \exp\left(\left.\frac \zeta_N(s,w \mid a_1, \ldots, a_N) \_ \right)\ , where \zeta_N is the Barnes zeta function. (This differs by a constant from Barnes's original definition.) Properties Considered as a meromorphic function of w, \Gamma_N(w\mid a_1,\ldots,a_N) has no zeros. It has poles at w= -\sum_^N n_ia_i for non-negative integers n_i. These poles are simple unless some of them coincide. Up to multiplication by the exponential of a polynomial, \Gamma_N(w\mid a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plot Of The Barnes G Aka Double Gamma Function G(z) 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 story 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 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 area Land * Plot (land), a piece of land used for building on ** Burial plot, a piece of land a person is buried in * Quadrat, a ... [...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] |
Q-gamma Function
In q-analog theory, the q-gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by . It is given by \Gamma_q(x) = (1-q)^\prod_^\infty \frac=(1-q)^\,\frac when , q, 1. Here (\cdot;\cdot)_\infty is the infinite q-Pochhammer symbol. The q-gamma function satisfies the functional equation \Gamma_q(x+1) = \frac\Gamma_q(x)= q\Gamma_q(x) In addition, the q-gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey (). For non-negative integers ''n'', \Gamma_q(n)= -1q! where cdotq is the q-factorial function. Thus the q-gamma function can be considered as an extension of the q-factorial function to the real numbers. The relation to the ordinary gamma function is made explicit in the limit \lim_ \Gamma_q(x) = \Gamma(x). There is a simple proof of this limit by Gosper. See the appendix of (). Transformation properties The q-gamma function satisfies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Gamma Function
In mathematics, the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely related to a function studied by , and can be expressed in terms of the triple gamma function. It is given by :\Gamma (z;p,q) = \prod_^\infty \prod_^\infty \frac. It obeys several identities: :\Gamma(z;p,q)=\frac\, :\Gamma(pz;p,q)=\theta (z;q) \Gamma (z; p,q)\, and :\Gamma(qz;p,q)=\theta (z;p) \Gamma (z; p,q)\, where θ is the q-theta function. When p=0, it essentially reduces to the infinite q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ...: :\Gamma(z;0,q)=\frac. Multiplication Formula Define :\tilde(z;p,q):=\frac(\theta(q;p))^\prod_^\infty \prod_^\infty \frac. Then the following formula hol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barnes Zeta Function
In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by . It is further generalized by the Shintani zeta function. Definition The Barnes zeta function is defined by : \zeta_N(s,w\mid a_1,\ldots,a_N)=\sum_\frac where ''w'' and ''a''''j'' have positive real part and ''s'' has real part greater than ''N''. It has a meromorphic continuation to all complex ''s'', whose only singularities are simple poles at ''s'' = 1, 2, ..., ''N''. For ''N'' = ''w'' = ''a''1 = 1 it is the Riemann zeta function. References * * * * * Zeta and L-functions {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meromorphic Function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are pole (complex analysis), poles of the function. The term comes from the Greek ''meros'' ( μέρος), meaning "part". Every meromorphic function on ''D'' can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on ''D'': any pole must coincide with a zero of the denominator. Heuristic description Intuitively, a meromorphic function is a ratio of two well-behaved (holomorphic) functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. If the denominator has a zero at ''z'' and the numerator does not, then the value of the function will approach infinity; if both parts have a zero at ''z'', then one must compare the multiplicity of these zero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-dimensional Conformal Field Theory
A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal field theories have infinite-dimensional symmetry algebras. In some cases, this allows them to be solved exactly, using the conformal bootstrap method. Notable two-dimensional conformal field theories include minimal models, Liouville theory, massless free bosonic theories, Wess–Zumino–Witten models, and certain sigma models. Basic structures Geometry Two-dimensional conformal field theories (CFTs) are defined on Riemann surfaces, where local conformal maps are holomorphic functions. While a CFT might conceivably exist only on a given Riemann surface, its existence on any surface other than the sphere implies its existence on all surfaces. Given a CFT, it is indeed possible to glue two Riemann surfaces where it exists, and obtain t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virasoro Algebra
In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. Definition The Virasoro algebra is spanned by generators for and the central charge . These generators satisfy ,L_n0 and The factor of 1/12 is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro algebra. The Virasoro algebra has a presentation in terms of two generators (e.g. 3 and −2) and six relations. Representation theory Highest weight representations A highest weight representation of the Virasoro algebra is a representation generated by a primary state: a vector v such that : L_ v = 0, \quad L_0 v = hv, where the number is called the conformal dimension or conformal weight of v.P. Di Francesco, P. Mathieu, and D. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville Field Theory
In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation. Liouville theory is defined for all complex values of the central charge c of its Virasoro symmetry algebra, but it is unitary only if :c\in(1,+\infty), and its classical limit is : c\to +\infty. Although it is an interacting theory with a continuous spectrum, Liouville theory has been solved. In particular, its three-point function on the sphere has been determined analytically. Introduction Liouville theory describes the dynamics of a field \phi called the Liouville field, which is defined on a two-dimensional space. This field is not a free field due to the presence of an exponential potential : V(\phi) = e^\ , where the parameter b is called the coupling constant. In a free field theory, the energy eigenvectors e^ are linearly independent, and the momentum \alpha is conserved in intera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
