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Generalized Polygamma Function
In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll. It generalizes the polygamma function to negative and fractional order, but remains equal to it for integer positive orders. Definition The generalized polygamma function is defined as follows: : \psi(z,q)=\frac or alternatively, : \psi(z,q)=e^\frac\left(e^\frac\right), where is the Polygamma function and , is the Hurwitz zeta function. The function is balanced, in that it satisfies the conditions :f(0)=f(1) \quad \text \quad \int_0^1 f(x)\, dx = 0. Relations Several special functions can be expressed in terms of generalized polygamma function. :\begin \psi(x) &= \psi(0,x)\\ \psi^(x)&=\psi(n,x) \qquad n\in\mathbb \\ \Gamma(x)&=\exp\left( \psi(-1,x)+\tfrac12 \ln 2\pi \right)\\ \zeta(z,q)&=\frac \left(2^ \psi \left(z-1,\frac\right)+2^ \psi \left(z-1,\frac\right)-\psi(z-1,q)\right)\\ \zeta'(-1,x)&=\psi(-2, x) + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Hugo Moll
Victor Hugo Moll (born 1956) is a Chilean American mathematician specializing in calculus. Moll studied at the Universidad Santa Maria and at the New York University with a master's degree in 1982 and a doctorate in 1984 with Henry P. McKean (''Stability in the Large for Solitary Wave Solutions to McKean's Nerve Conduction Caricature''). He was a post-doctoral student at Temple University and became an assistant professor in 1986 and an associate professor in 1992 and in 2001 Professor at Tulane University. In 1990–1991, he was a visiting professor at the University of Utah, in 1999 at the Universidad Técnica Federico Santa María in Valparaíso, and in 1995 a visiting scientist at the Courant Institute of Mathematical Sciences of New York University. He deals with classical analysis, symbolic arithmetic and experimental mathematics, special functions and number theory. Projects Inspired by a 1988 paper in which proved several integrals in '' Table of Integrals, Series ... [...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] |
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] |
Hurwitz Zeta Function
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables with and by :\zeta(s,a) = \sum_^\infty \frac. This series is absolutely convergent for the given values of and and can be extended to a meromorphic function defined for all . The Riemann zeta function is . The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882. Integral representation The Hurwitz zeta function has an integral representation :\zeta(s,a) = \frac \int_0^\infty \frac dx for \operatorname(s)>1 and \operatorname(a)>0. (This integral can be viewed as a Mellin transform.) The formula can be obtained, roughly, by writing :\zeta(s,a)\Gamma(s) = \sum_^\infty \frac \int_0^\infty x^s e^ \frac = \sum_^\infty \int_0^\infty y^s e^ \frac and then interchanging the sum and integral. The integral representation above can be converted to a contour integral representation :\zeta(s,a) = -\Gamma(1-s)\frac \int_C \frac dz where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Polynomials
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in the study of many special functions and, in particular, the Riemann zeta function and the Hurwitz zeta function. They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). For the Bernoulli polynomials, the number of crossings of the ''x''-axis in the unit interval does not go up with the degree. In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions. A similar set of polynomials, based on a generating function, is the family of Euler polynomials. Representations The Bernoulli polynomials ''B''''n'' can be defined by a generating function. They also admit a variety of derived representations. Generating functions The generating function for the Bernoulli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-function
In mathematics, the -function, typically denoted ''K''(''z''), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the gamma function. Definition Formally, the -function is defined as :K(z)=(2\pi)^ \exp\left binom+\int_0^ \ln \Gamma(t + 1)\,dt\right It can also be given in closed form as :K(z)=\exp\bigl zeta'(-1,z)-\zeta'(-1)\bigr/math> where denotes the derivative of the Riemann zeta function, denotes the Hurwitz zeta function and :\zeta'(a,z)\ \stackrel\ \left.\frac\_. Another expression using the polygamma function is :K(z)=\exp\left psi^(z)+\frac-\frac \ln 2\pi \right/math> Or using the balanced generalization of the polygamma function: :K(z)=A \exp\left psi(-2,z)+\frac\right/math> where is the Glaisher constant. Similar to the Bohr-Mollerup Theorem for the Gamma function, the log K-function is the unique (up to an additive constant) eventually 2-convex solution to the equation \Delta f(x)=x\ln(x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glaisher Constant
Glaisher is a surname, and may refer to: *Cecilia Glaisher (1828–1892), photographer and illustrator * James Glaisher (1809–1903), English meteorologist and astronomer * James Whitbread Lee Glaisher (1848–1928), English mathematician and astronomer See also *Glaisher (crater) Glaisher is a lunar impact crater that is located in the region of terrain that forms the southwest border of Mare Crisium. It lies to the southwest of the lava-flooded crater Yerkes, and west-northwest of the Greaves–Lick crater pair. It is s ..., a crater on the moon, named for James Glaisher (1809–1903) * Glacier (other) {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Constant
In mathematics, Catalan's constant , is defined by : G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots, where is the Dirichlet beta function. Its numerical value is approximately : It is not known whether is irrational, let alone transcendental. has been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven". Catalan's constant was named after Eugène Charles Catalan, who found quickly-converging series for its calculation and published a memoir on it in 1865. Uses In low-dimensional topology, Catalan's constant is 1/4 of the volume of an ideal hyperbolic octahedron, and therefore 1/4 of the hyperbolic volume of the complement of the Whitehead link. It is 1/8 of the volume of the complement of the Borromean rings. In combinatorics and statistical mechanics, it arises in connection with counting domino tilings, spanning trees, and Hamiltonian cycles of grid graphs. In number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
