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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 ... [...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 fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperfactorial
In mathematics, and more specifically number theory, the hyperfactorial of a positive integer n is the product of the numbers of the form x^x from 1^1 to n^n. Definition The hyperfactorial of a positive integer n is the product of the numbers 1^1, 2^2, \dots n^n. That is, H(n) = 1^1\cdot 2^2\cdot \cdots n^n = \prod_^ i^i = n^n H(n-1). Following the usual convention for the empty product, the hyperfactorial of 0 is 1. The integer sequence of hyperfactorials, beginning with H(0)=1, is: Interpolation and approximation The hyperfactorials were studied beginning in the 19th century by Hermann Kinkelin and James Whitbread Lee Glaisher. As Kinkelin showed, just as the factorials can be continuously interpolated by the gamma function, the hyperfactorials can be continuously interpolated by the K-function. Glaisher provided an asymptotic formula for the hyperfactorials, analogous to Stirling's formula for the factorials: H(n)=An^e^\left(1+\frac-\frac+\cdots\right), where A\approx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with re ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the deriv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ( zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > 1 and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article " On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is ... [...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 w ... [...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 fo ... [...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] |
