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Nørlund–Rice Integral
In mathematics, the Nørlund–Rice integral, sometimes called Rice's method, relates the ''n''th forward difference of a function to a contour integral on the complex plane. It commonly appears in the theory of finite differences and has also been applied in computer science and graph theory to estimate binary tree lengths. It is named in honour of Niels Erik Nørlund and Stephen O. Rice. Nørlund's contribution was to define the integral; Rice's contribution was to demonstrate its utility by applying saddle-point techniques to its evaluation. Definition The ''n''th forward difference of a function ''f''(''x'') is given by :\Delta^n x)= \sum_^n (-1)^ f(x+k) where is the binomial coefficient. The Nørlund–Rice integral is given by :\sum_^n (-1)^ f(k) = \frac \oint_\gamma \frac\, dz where ''f'' is understood to be meromorphic, α is an integer, 0\leq \alpha \leq n, and the contour of integration is understood to circle the poles located at the integers α, ..., ''n'', ... [...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] |
Binomial Transform
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function. Definition The binomial transform, , of a sequence, , is the sequence defined by s_n = \sum_^n (-1)^k \binom a_k. Formally, one may write s_n = (Ta)_n = \sum_^n T_ a_k for the transformation, where is an infinite-dimensional operator with matrix elements . The transform is an involution, that is, TT = 1 or, using index notation, \sum_^\infty T_ T_ = \delta_ where \delta_ is the Kronecker delta. The original series can be regained by a_n=\sum_^n (-1)^k \binom s_k. The binomial transform of a sequence is just the -th forward differences of the sequence, with odd differences carrying a negative sign, namely: \begin s_0 &= a_0 \\ s_1 &= - (\Delta a)_0 = -a_1+a_0 \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, '' holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which dea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial And Binomial Topics
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 function an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Art Of Computer Programming
''The Art of Computer Programming'' (''TAOCP'') is a comprehensive multi-volume monograph written by the computer scientist Donald Knuth presenting programming algorithms and their analysis. it consists of published volumes 1, 2, 3, 4A, and 4B, with more expected to be released in the future. The Volumes 1–5 are intended to represent the central core of computer programming for sequential machines; the subjects of Volumes 6 and 7 are important but more specialized. When Knuth began the project in 1962, he originally conceived of it as a single book with twelve chapters. The first three volumes of what was then expected to be a seven-volume set were published in 1968, 1969, and 1973. Work began in earnest on Volume 4 in 1973, but was suspended in 1977 for work on typesetting prompted by the second edition of Volume 2. Writing of the final copy of Volume 4A began in longhand in 2001, and the first online pre-fascicle, 2A, appeared later in 2001. The first published installment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Factorial And Binomial Topics
{{Short description, none This is a list of factorial and binomial topics in mathematics. See also binomial (other). * Abel's binomial theorem *Alternating factorial *Antichain *Beta function * Bhargava factorial *Binomial coefficient **Pascal's triangle *Binomial distribution *Binomial proportion confidence interval * Binomial-QMF ( Daubechies wavelet filters) *Binomial series *Binomial theorem *Binomial transform *Binomial type * Carlson's theorem *Catalan number ** Fuss–Catalan number *Central binomial coefficient *Combination *Combinatorial number system *De Polignac's formula *Difference operator * Difference polynomials *Digamma function * Egorychev method * Erdős–Ko–Rado theorem *Euler–Mascheroni constant *Faà di Bruno's formula *Factorial * Factorial moment *Factorial number system *Factorial prime * Factoriangular number *Gamma distribution *Gamma function *Gaussian binomial coefficient * Gould's sequence * Hyperfactorial *Hypergeometric distribution * Hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Table Of Newtonian Series
In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a_n written in the form :f(s) = \sum_^\infty (-1)^n a_n = \sum_^\infty \frac a_n where : is the binomial coefficient and (s)_n is the rising factorial, falling factorial. Newtonian series often appear in relations of the form seen in umbral calculus. List The generalized binomial theorem gives : (1+z)^s = \sum_^z^n = 1+z+z^2+\cdots. A proof for this identity can be obtained by showing that it satisfies the differential equation : (1+z) \frac = s (1+z)^s. The digamma function: :\psi(s+1)=-\gamma-\sum_^\infty \frac . The Stirling numbers of the second kind are given by the finite sum :\left\ =\frac\sum_^(-1)^ j^n. This formula is a special case of the ''k''th forward difference of the monomial ''x''''n'' evaluated at ''x'' = 0: : \Delta^k x^n = \sum_^(-1)^ (x+j)^n. A related identity forms the basis of the Nörlund–Rice integral: :\sum_^n \frac = \frac = \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function. The theory of asymptotic series was created by Poincaré (and independently by Stieltjes) in 1886. The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. Since a '' convergent'' Taylor s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perron's Formula
In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform. Statement Let \ be an arithmetic function, and let : g(s)=\sum_^ \frac be the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for \Re(s)>\sigma. Then Perron's formula is : A(x) = ' a(n) =\frac\int_^ g(z)\frac \,dz. Here, the prime on the summation indicates that the last term of the sum must be multiplied by 1/2 when ''x'' is an integer. The integral is not a convergent Lebesgue integral; it is understood as the Cauchy principal value. The formula requires that ''c'' > 0, ''c'' > σ, and ''x'' > 0. Proof An easy sketch of the proof comes from taking Abel's sum formula : g(s)=\sum_^ \frac=s\int_^ A(x)x^ dx. This is nothing but a Laplace transform under the variable change x = e^t. Inverting it one gets Perron's formula. Examples Be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riesz Mean
In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean. The Riesz mean should not be confused with the Bochner–Riesz mean or the Strong–Riesz mean. Definition Given a series \, the Riesz mean of the series is defined by :s^\delta(\lambda) = \sum_ \left(1-\frac\right)^\delta s_n Sometimes, a generalized Riesz mean is defined as :R_n = \frac \sum_^n (\lambda_k-\lambda_)^\delta s_k Here, the \lambda_n are a sequence with \lambda_n\to\infty and with \lambda_/\lambda_n\to 1 as n\to\infty. Other than this, the \lambda_n are taken as arbitrary. Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the case of s_n = \sum_^n a_k for some sequence \. Typically, a sequence is summable when the limit \lim_ R_n exists, or the limit \lim_s^\delta(\lambda) exists, although the precise summability theorems in question often impose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan's Master Theorem
In mathematics, Ramanujan's master theorem, named after Srinivasa Ramanujan, is a technique that provides an analytic expression for the Mellin transform of an analytic function. The result is stated as follows: If a complex-valued function f(x) has an expansion of the form f(x)=\sum_^\infty \frac(-x)^k then the Mellin transform of f(x) is given by \int_0^\infty x^ f(x) \, dx = \Gamma(s)\,\varphi(-s) where \Gamma(s) is the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series. Higher-dimensional versions of this theorem also appear in quantum physics through Feynman diagrams. A similar result was also obtained by Glaisher. Alternative formalism An alternative formulation of Ramanujan's master theorem is as follows: \int_0^\infty x^\left(\,\lambda(0) - x\,\lambda(1) + x^2\,\lambda(2) -\,\cdots\,\right) dx = \frac\,\lambda(-s) which gets converted to the above form after substituting \lambda(n) \equiv \frac and us ... [...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] |
