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Euler–Boole Summation
Euler–Boole summation is a method for summing alternating series based on Euler's polynomials, which are defined by : \frac=\sum_^\infty E_n(x)\frac. The concept is named after Leonhard Euler and George Boole. The periodic Euler functions are :\widetilde E_n(x+1)=-\widetilde E_n(x)\text \widetilde E_n(x)=E_n(x) \text 0 |
Alternating Series
In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. Examples The geometric series 1/2 − 1/4 %2B 1/8 − 1/16 %2B %E2%8B%AF sums to 1/3. The alternating harmonic series has a finite sum but the harmonic series does not. The Mercator series provides an analytic expression of the natural logarithm: \sum_^\infty \frac x^n \;=\; \ln (1+x). The functions sine and cosine used in trigonometry can be defined as alternating series in calculus even though they are introduced in elementary algebra as the ratio of sides of a right triangle. In fact, \sin x = \sum_^\infty (-1)^n \frac, and \cos x = \sum_^\infty (-1)^n \frac . When the alternating factor is removed from these series one obtains the hyperbolic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's 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 Berno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. He worked in the fields of differential equations and algebraic logic, and is best known as the author of ''The Laws of Thought'' (1854) which contains Boolean algebra. Boolean logic is credited with laying the foundations for the Information Age. Early life Boole was born in 1815 in Lincoln, Lincolnshire, England, the son of John Boole senior (1779–1848), a shoemaker and Mary Ann Joyce. He had a primary school education, and received lessons from his father, but due to a serious decline in business, he had little further formal and academic teaching. William Brooke, a bookseller in Lincoln, may have helped him with Latin, which he may also have learned at the school of Thomas Bainbridge. He was self-taught in modern languages.H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neil J
Neil is a masculine name of Gaelic and Irish origin. The name is an anglicisation of the Irish ''Niall'' which is of disputed derivation. The Irish name may be derived from words meaning "cloud", "passionate", "victory", "honour" or "champion".. As a surname, Neil is traced back to Niall of the Nine Hostages who was an Irish king and eponymous ancestor of the Uí Néill and MacNeil kindred. Most authorities cite the meaning of Neil in the context of a surname as meaning "champion". Origins The Gaelic name was adopted by the Vikings and taken to Iceland as ''Njáll'' (see Nigel). From Iceland it went via Norway, Denmark, and Normandy to England. The name also entered Northern England and Yorkshire directly from Ireland, and from Norwegian settlers. ''Neal'' or ''Neall'' is the Middle English form of ''Nigel''. As a first name, during the Middle Ages, the Gaelic name of Irish origins was popular in Ireland and later Scotland. During the 20th century ''Neil'' began to be used in Engl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
