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Leibniz Formula For π
In mathematics, the Leibniz formula for , named after Gottfried Leibniz, states that 1-\frac+\frac-\frac+\frac-\cdots=\frac, an alternating series. It is also called the Madhava–Leibniz series as it is a special case of a more general series expansion for the inverse tangent function, first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century, the specific case first published by Leibniz around 1676. The series for the inverse tangent function, which is also known as Gregory's series, can be given by: : \arctan x = x - \frac + \frac - \frac + \cdots The Leibniz formula for \frac can be obtained by putting x=1 into this series. It also is the Dirichlet -series of the non-principal Dirichlet character of modulus 4 evaluated at s=1, and, therefore, the value of the Dirichlet beta function. Proofs Proof 1 \begin \frac &= \arctan(1) \\ &= \int_0^1 \frac 1 \, dx \\ pt& = \int_0^1\left(\sum_^n (-1)^k x^+\frac\right) \, dx \\ pt& = \left(\sum ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shanks Transformation
In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941.Weniger (2003). Formulation For a sequence \left\_ the series :A = \sum_^\infty a_m\, is to be determined. First, the partial sum A_n is defined as: :A_n = \sum_^n a_m\, and forms a new sequence \left\_. Provided the series converges, A_n will also approach the limit A as n\to\infty. The Shanks transformation S(A_n) of the sequence A_n is the new sequence defined byBender & Orszag (1999), pp. 368–375.Van Dyke (1975), pp. 202–205. :S(A_n) = \frac = A_ - \frac where this sequence S(A_n) often converges more rapidly than the sequence A_n. Further speed-up may be obtained by repeated use of the Shanks transformation, by computing S^2(A_n)=S(S(A_n)), S^3(A_n)=S(S(S(A_n))), etc. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which alway ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Product
In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a''''n'' as ''n'' increases without bound. The product is said to '' converge'' when the limit exists and is not zero. Otherwise the product is said to ''diverge''. A limit of zero is treated specially in order to obtain results analogous to those for infinite sums. Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence ''a''''n'' as ''n'' increases without bound must be 1, while the converse is in general not true. The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète ( Viète's formula, the first p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Series
In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet. Combinatorial importance Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian products. Suppose that ''A'' is a set with a function ''w'': ''A'' → N assigning a weight to each of the elements of ''A'', and suppose additionally that the fibre over any natural number under that weight is a finite set. (We call such an arrangement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jonathan Borwein
Jonathan Michael Borwein (20 May 1951 – 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. He was a close associate of David H. Bailey, and they have been prominent public advocates of experimental mathematics. Borwein's interests spanned pure mathematics (analysis), applied mathematics (optimization), computational mathematics (numerical and computational analysis), and high performance computing. He authored ten books, including several on experimental mathematics, a monograph on convex functions, and over 400 refereed articles. He was a co-founder in 1995 of software company MathResources, consulting and producing interactive software primarily for school and university mathematics. Borwein was also an expert on the number pi and especially its computation. Early life and education Borwein was born in St. Andrews, Scotland in 1951 into a Jewish family. His father was mat ... [...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 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 series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Number
In mathematics, the Euler numbers are a sequence ''En'' of integers defined by the Taylor series expansion :\frac = \frac = \sum_^\infty \frac \cdot t^n, where \cosh (t) is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely: :E_n=2^nE_n(\tfrac 12). The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements. Examples The odd-indexed Euler numbers are all zero. The even-indexed ones have alternating signs. Some values are: : Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive . This article adheres to the convention adopted above. Explicit formulas In terms of Stirling numbers of the second kind Follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decimal Representation
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, is a nonnegative integer, and b_0, \ldots, b_k, a_1, a_2,\ldots are ''digits'', which are symbols representing integers in the range 0, ..., 9. Commonly, b_k\neq 0 if k > 1. The sequence of the a_i—the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0. If all a_i are , the separator is also omitted, resulting in a finite sequence of digits, which represents a natural number. The decimal representation represents the infinite sum: r=\sum_^k b_i 10^i + \sum_^\infty \frac. Every nonnegative real number has at least one such representation; it has two such representations (with b_k\neq 0 if k>0) if and only if one has a trailing infinite sequence of , and the other has a trailing infin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to ''quadrature'') is more or less a synonym for ''numerical integration'', especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take ''quadrature'' to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel–Plana Formula
In mathematics, the Abel–Plana formula is a summation formula discovered independently by and . It states that :\sum_^\infty f(n)=\frac 1 2 f(0)+ \int_0^\infty f(x) \, dx+ i \int_0^\infty \frac \, dt. It holds for functions ''f'' that are holomorphic in the region Re(''z'') ≥ 0, and satisfy a suitable growth condition in this region; for example it is enough to assume that , ''f'', is bounded by ''C''/, ''z'', 1+ε in this region for some constants ''C'', ε > 0, though the formula also holds under much weaker bounds. . An example is provided by the Hurwitz zeta function, :\zeta(s,\alpha)= \sum_^\infty \frac = \frac + \frac 1 + 2\int_0^\infty\frac\frac, which holds for all s \in \mathbb, . Abel also gave the following variation for alternating sums: :\sum_^\infty (-1)^nf(n)= \frac f(0)+i \int_0^\infty \frac \, dt. Which is related to the Lindelöf summation formula \sum_^(-1)^kf(k)=(-1)^m\int_^f(m-1/2+ix)\frac Proof Let f be holomorphic on \Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Maclaurin Formula
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence. The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals. It was later generalized to Darboux's formula. The formula If and are natural numbers and is a real or complex valued continuous function for real numbers in the interval , then the integral I = \int_m^n f(x)\,dx can be approximated by the sum (or vice versa) S = f(m + 1) + \cdots + f(n - 1) + f(n) (see rectangle method). The Euler–Maclaurin form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
