Zeldovich regularization refers to a regularization method to calculate divergent integrals and divergent series, that was first introduced by Yakov Zeldovich in 1961. [1] Zeldovich was originally interested in calculating the norm of the Gamow wave function which is divergent since there is an outgoing spherical wave. Zeldovich regularization uses a Gaussian type-regularization and is defined, for divergent integrals, by [2]
and, for divergent series, by [3] [4] [5]
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation for an arbitrary complex number , which represents the order of the Bessel function. Although and produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of .
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. 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 in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.
In mathematics, the Laplace transform, named after Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable .
In mathematical analysis, the Dirac delta function, also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions is the product of their Fourier transforms. More generally, convolution in one domain equals point-wise multiplication in the other domain. Other versions of the convolution theorem are applicable to various Fourier-related transforms.
In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals, this typically involves unboundedness, either of the set over which the integral is taken or of the integrand, or both. It may also involve bounded but not closed sets or bounded but not continuous functions. While an improper integral is typically written symbolically just like a standard definite integral, it actually represents a limit of a definite integral or a sum of such limits; thus improper integrals are said to converge or diverge. If a regular definite integral is worked out as if it is improper, the same answer will result.
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is
In physics, the S-matrix or scattering matrix is a matrix which relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
In mathematical analysis, Cesàro summation assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the 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.
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 best possible sum for such series. There are several variations of this method that are also called Borel summation, and a generalization of it called Mittag-Leffler summation.
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form where and the integrands are functions dependent on the derivative of this integral is expressible as where the partial derivative indicates that inside the integral, only the variation of with is considered in taking the derivative.
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 be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory.
The Fransén–Robinson constant, sometimes denoted F, is the mathematical constant that represents the area between the graph of the reciprocal Gamma function, 1/Γ(x), and the positive x axis. That is,
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 sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.
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
In mathematics, the Abel–Plana formula is a summation formula discovered independently by Niels Henrik Abel and Giovanni Antonio Amedeo Plana. It states that
In potential theory, a branch of mathematics, the Laplacian of the indicator of the domain D is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the surface of D. It can be viewed as the surface delta prime function. It is analogous to the second derivative of the Heaviside step function in one dimension. It can be obtained by letting the Laplace operator work on the indicator function of some domain D.
