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Abel's summation formula, the Glossary

In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series.^[1]

21 relations: Abel–Plana formula, Analytic number theory, Complex number, Dirichlet series, Function (mathematics), Harmonic number, Integration by parts, Mathematics, Möbius function, Mertens function, Niels Henrik Abel, Real number, Residue (complex analysis), Riemann zeta function, Riemann–Stieltjes integral, Sequence, Series (mathematics), Special functions, Summation by parts, Undergraduate Texts in Mathematics, Zeros and poles.
Summability methods

Abel–Plana formula

In mathematics, the Abel–Plana formula is a summation formula discovered independently by and. Abel's summation formula and Abel–Plana formula are Summability methods.

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In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. Abel's summation formula and analytic number theory are number theory.

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Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

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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.

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Function (mathematics)

In mathematics, a function from a set to a set assigns to each element of exactly one element of.

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Harmonic number

In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n. Abel's summation formula and harmonic number are number theory.

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Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Möbius function

The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832.

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Mertens function

In number theory, the Mertens function is defined for all positive integers n as where \mu(k) is the Möbius function.

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Niels Henrik Abel

Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields.

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Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

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Residue (complex analysis)

In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities.

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Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s).

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Riemann–Stieltjes integral

In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes.

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Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters.

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Series (mathematics)

In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

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Special functions

Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.

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Summation by parts

In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. Abel's summation formula and summation by parts are Summability methods.

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