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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 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= B(n+1,s-n),s ... [...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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^(1-t)^\,dt for complex number inputs z_1, z_2 such that \Re(z_1), \Re(z_2)>0. The beta function was studied by Leonhard Euler and Adrien-Marie Legendre and was given its name by Jacques Binet; its symbol is a Greek capital beta. Properties The beta function is symmetric, meaning that \Beta(z_1,z_2) = \Beta(z_2,z_1) for all inputs z_1 and z_2.Davis (1972) 6.2.2 p.258 A key property of the beta function is its close relationship to the gamma function: : \Beta(z_1,z_2)=\frac. A proof is given below in . The beta function is also closely related to binomial coefficients. When (or , by symmetry) is a positive integer, it follows from the definition of the gamma function thatDavis (1972) 6.2.1 p.258 : \Beta(m,n) =\dfrac = \frac \B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Differences
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted \Delta is the operator that maps a function to the function \Delta /math> defined by :\Delta x)= f(x+1)-f(x). A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations, specially in the solving methods. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for approximating derivatives, and the term "fini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlson's Theorem
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers. The theorem may be obtained from the Phragmén–Lindelöf theorem, which is itself an extension of the maximum-modulus theorem. Carlson's theorem is typically invoked to defend the uniqueness of a Newton series expansion. Carlson's theorem has generalized analogues for other expansions. Statement Assume that satisfies the following three conditions: the first two conditions bound the growth of at infinity, whereas the third one states that vanishes on the non-negative integers. * is an entire function of exponential type, meaning that , f(z), \leq C e^, \quad z \in \mathbb for some real values , . * There exists such that , f(iy), \leq C e^, \quad y \in \mathbb * for any non-negative integer . Then i ... [...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 *Gamma distribution *Gamma function *Gaussian binomial coefficient *Gould's sequence *Hyperfactorial *Hypergeometric distribution * Hypergeometric function identities ... [...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, ''T'', of a sequence, , is the sequence defined by :s_n = \sum_^n (-1)^k a_k. Formally, one may write :s_n = (Ta)_n = \sum_^n T_ a_k for the transformation, where ''T'' is an infinite-dimensional operator with matrix elements ''T''''nk''. 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 s_k. The binomial transform of a sequence is just the ''n''th forward differences of the sequence, with odd differences carrying a negative sign, namely: :\begin s_0 &= a_0 \\ s_1 &= - (\Delta a) ... [...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] |
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 Numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Number Theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. *Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Falling Factorial
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \end The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as :\begin x^ = x^\overline &= \overbrace^ \\ &= \prod_^n(x+k-1) = \prod_^(x+k) \,. \end The value of each is taken to be 1 (an empty product) when . These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation , where is a non-negative integer. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used with yet another meaning, namely to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
