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Dirichlet Eta Function
In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: \eta(s) = \sum_^ = \frac - \frac + \frac - \frac + \cdots\approx \prod_^ \infty . This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ''ζ''(''s'') — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ''ζ''*(''s''). The following relation holds: \eta(s) = \left(1-2^\right) \zeta(s) Both Dirichlet eta function and Riemann zeta function are special cases of Polylogarithm. While the Dirichlet series expansion for the eta function is convergent only for any complex number ''s'' with real part > 0, it is Abel summable for any complex number. This serves to define the eta function as an entire function. (The above relation and the facts that the eta function is ent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Eta Function
In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: \eta(s) = \sum_^ = \frac - \frac + \frac - \frac + \cdots\approx \prod_^ \infty . This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ''ζ''(''s'') — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ''ζ''*(''s''). The following relation holds: \eta(s) = \left(1-2^\right) \zeta(s) Both Dirichlet eta function and Riemann zeta function are special cases of Polylogarithm. While the Dirichlet series expansion for the eta function is convergent only for any complex number ''s'' with real part > 0, it is Abel summable for any complex number. This serves to define the eta function as an entire function. (The above relation and the facts that the eta function is ent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Widder
David Vernon Widder (25 March 1898 – 8 July 1990) was an American mathematician. He earned his Ph.D. at Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ... in 1924 under George Birkhoff and went on to join the faculty there. He was a co-founder of the '' Duke Mathematical Journal'' and the author of the textbook ''Advanced Calculus''. He wrote also '' The Laplace transform'' (in which he gave a first solution to Landau's problem on the Dirichlet eta function), ''An introduction to transform theory'', and '' The convolution transform'' (co-author with I. I. Hirschman). References *''A Century of Mathematics in America'' by Peter L. Duren and Richard Askey, American Mathematical Society, 1988, . *''A History of the Second Fifty Years, American Mathematical So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Number
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 ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 − 2 + 3 − 4 + · · ·
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grandi's Series
In mathematics, the infinite series , also written : \sum_^\infty (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it lacks a sum in the usual sense. On the other hand, its Cesàro sum is 1/2. Unrigorous methods One obvious method to attack the series :1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + ... is to treat it like a telescoping series and perform the subtractions in place: :(1 − 1) + (1 − 1) + (1 − 1) + ... = 0 + 0 + 0 + ... = 0. On the other hand, a similar bracketing procedure leads to the apparently contradictory result :1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + ... = 1 + 0 + 0 + 0 + ... = 1. Thus, by applying parentheses to Grandi's series in different ways, one can obtain either 0 or 1 as a "value". (Variations of this idea, called the Eilenberg–Mazur swindle, are sometimes used in knot theory and algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel Summation
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series :1 + \frac + \frac + \frac + \frac + \cdots =\sum_^\infty\frac. The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A ''summability method'' or ''summation method'' is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergent se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Mathematical Society
The Canadian Mathematical Society (CMS) (french: Société mathématique du Canada) is an association of professional mathematicians dedicated to the interests of mathematical research, outreach, scholarship and education in Canada. It serves the national community through the publication of academic journals, community bulletins, and the administration of mathematical competitions. It was originally conceived in June 1945 as the Canadian Mathematical Congress. A name change was debated for many years; ultimately, a new name was adopted in 1979, upon its incorporation as a non-profit charitable organization. The society is also affiliated with various national and international mathematical societies, including the Canadian Applied and Industrial Mathematics Society and the Society for Industrial and Applied Mathematics. The society is also a member of the International Mathematical Union and the International Council for Industrial and Applied Mathematics. History The Canadian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Polynomials
The Chebyshev polynomials are two sequences of polynomials related to the trigonometric functions, cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind T_n are defined by : T_n(\cos \theta) = \cos(n\theta). Similarly, the Chebyshev polynomials of the second kind U_n are defined by : U_n(\cos \theta) \sin \theta = \sin\big((n + 1)\theta\big). That these expressions define polynomials in \cos\theta may not be obvious at first sight, but follows by rewriting \cos(n\theta) and \sin\big((n+1)\theta\big) using de Moivre's formula or by using the List of trigonometric identities#Angle sum and difference identities, angle sum formulas for \cos and \sin repeatedly. For example, the List of trigonometric identities#Double-angle formulae, double angle formulas, which follow directly from the angle sum formulas, may be used to obtain T_2(\cos\t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Borwein
Peter Benjamin Borwein (born St. Andrews, Scotland, May 10, 1953 – 23 August 2020) was a Canadian mathematician and a professor at Simon Fraser University. He is known as a co-author of the paper which presented the Bailey–Borwein–Plouffe algorithm (discovered by Simon Plouffe) for computing π. First interest in mathematics Borwein was born into a Jewish family. He became interested in number theory and classical analysis during his second year of university. He had not previously been interested in math, although his father was the head of the University of Western Ontario's mathematics department and his mother is associate dean of medicine there. Borwein and his two siblings majored in mathematics. Academic career After completing a Bachelor of Science in Honours Math at the University of Western Ontario in 1974, he went on to complete an MSc and Ph.D. at the University of British Columbia. He joined the Department of Mathematics at Dalhousie University. Whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forward Difference
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] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
