Category: Summability methods

In mathematics and theoretical physics, resummation is a procedure to obtain a finite result from a divergent sum (series) of functions. Resummation involves a definition of another (convergent) funct
Regularization (physics)
In physics, especially quantum field theory, regularization is a method of modifying observables which have singularities in order to make them finite by the introduction of a suitable parameter calle
Darboux's formula
In mathematical analysis, Darboux's formula is a formula introduced by Gaston Darboux for summing infinite series by using integrals or evaluating integrals using infinite series. It is a generalizati
Series acceleration
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numeri
Abel's theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.
Hadamard regularization
In mathematics, Hadamard regularization (also called Hadamard finite part or Hadamard's partie finie) is a method of regularizing divergent integrals by dropping some divergent terms and keeping the f
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. It is
Divisor sum identities
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or
Euler–Boole summation
Euler–Boole summation is a method for summing alternating series based on Euler's polynomials, which are defined by The concept is named after Leonhard Euler and George Boole. The periodic Euler funct
Borel summation
In mathematics, Borel summation is a summation method for divergent series, introduced by Émile Borel. It is particularly useful for summing divergent asymptotic series, and in some sense gives the be
Lambert summation
In mathematical analysis, Lambert summation is a summability method for a class of divergent series.
Poisson summation formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transfo
Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
Euler summation
In the mathematics of convergent and divergent series, Euler summation is a summability method. That is, it is a method for assigning a value to a series, different from the conventional method of tak
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 eval
Mittag-Leffler summation
In mathematics, Mittag-Leffler summation is any of several variations of the Borel summation method for summing possibly divergent formal power series, introduced by Mittag-Leffler
Riesz mean
In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean. The Riesz mean should not be confused wi
Nachbin's theorem
In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article provi
Bochner–Riesz mean
The Bochner–Riesz mean is a summability method often used in harmonic analysis when considering convergence of Fourier series and Fourier integrals. It was introduced by Salomon Bochner as a modificat
Hölder summation
In mathematics, Hölder summation is a method for summing divergent series introduced by Hölder.
Abel–Plana formula
In mathematics, the Abel–Plana formula is a summation formula discovered independently by Niels Henrik Abel and Giovanni Antonio Amedeo Plana. It states that It holds for functions f that are holomorp
Ramanujan summation
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a
Zeta function regularization
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can b
Dimensional regularization
In theoretical physics, dimensional regularization is a method introduced by Giambiagi and as well as – independently and more comprehensively – by 't Hooft and Veltman for regularizing integrals in t
Divergent series
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 converge
Abelian and Tauberian theorems
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The
Perron's formula
In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transfo
Abel's summation formula
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.
Silverman–Toeplitz theorem
In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular. A regular matrix summabilit
Cesàro summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the