Analytic expressions for integrals of products of spherical Bessel functions

Mehrem, Rami; Londergan, J. T.; Macfarlane, M H

doi:10.1088/0305-4470/24/7/018

Cited by 49 publications

(53 citation statements)

References 10 publications

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“…These overlap integrals can be evaluated with the projection of vector TAM waves in Eq. (22). The angular parts of the overlap integral can be evaluated by writing the L/E/B vector spherical harmonics in terms of helicity spherical harmonics as…”

Section: Two Transverse Vectors and A Scalarmentioning

confidence: 99%

Wigner-Eckart theorem in cosmology: Bispectra for total-angular-momentum waves

Dai

Jeong²,

Kamionkowski

2013

Phys. Rev. D

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Total-angular-momentum (TAM) waves provide a set of basis functions for scalar, vector, and tensor fields that can be used in place of plane waves and that reflect the rotational symmetry of the spherical sky. Here we discuss three-point correlation functions, or bispectra in harmonic space, for scalar, vector, and tensor fields in terms of TAM waves. The Wigner-Eckart theorem dictates that the expectation value, assuming statistical isotropy, of the product of three TAM waves is the product of a Clebsch-Gordan coefficient (or Wigner-3j symbol) times a function only of the total-angular-momentum quantum numbers. Here we show how this works, and we provide explicit expressions relating the bispectra for TAM waves in terms of the more commonly used Fourier-space bispectra. This formalism will be useful to simplify calculations of projections of three-dimensional bispectra onto the spherical sky.

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Section: Two Transverse Vectors and A Scalarmentioning

confidence: 99%

Wigner-Eckart theorem in cosmology: Bispectra for total-angular-momentum waves

Dai

Jeong²,

Kamionkowski

2013

Phys. Rev. D

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“…On the other hand, there are many other applications in nuclear physics that encounter these types of integrals amongst which are distorted wave calculations [2] and nuclear response function calculations [3]. These types of integrals have in the past been calculated analytically (many references exist, see for example references [2] and [4][5][6][7][8][9][10][11]), or numerically using complex-plane methods (see for example references [12][13][14]). In this paper, an additional advantage to the plane wave expansion is presented, whereby the expansion itself is used to derive analytical solutions to integrals over spherical Bessel functions and identities involving them.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, an additional advantage to the plane wave expansion is presented, whereby the expansion itself is used to derive analytical solutions to integrals over spherical Bessel functions and identities involving them. The technique itself is not new [11], but the author is not aware of any prior paper that has highlighted its versatility. In section 2, the plane wave expansion is introduced from which the integral representation of the spherical Bessel function is derived.…”

Section: Introductionmentioning

confidence: 99%

The plane wave expansion, infinite integrals and identities involving spherical Bessel functions

Mehrem

2011

Applied Mathematics and Computation

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This paper shows that the plane wave expansion can be a useful tool in obtaining analytical solutions to infinite integrals over spherical Bessel functions and the derivation of identites for these functions. The integrals are often used in nuclear scattering calculations, where an analytical result can provide an insight into the reaction mechanism. A technique is developed whereby an integral over several special functions which cannot be found in any standard integral table can be broken down into integrals that have existing analytical solutions.

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“…Broda (2013) presented some results on truncated moments of the MGH distribution, extending the results of Imhof (1961), based on a numerical method of the inversion of the characteristic function. The results of our research complement Broda (2013), as the analytic expression we provide is based on the results on moments of the GIG distribution, that are functions of Bessel of the first and second kind, for which there exist analytic expressions such as in Mehrem et al (1991).…”

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confidence: 75%

Multivariate elliptical truncated moments

Arismendi

Broda

2017

Journal of Multivariate Analysis

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In this study, we derived analytic expressions for the elliptical truncated moment generating function (MGF), the zeroth-, first-, and second-order moments of quadratic forms of the multivariate normal, Student's t, and generalised hyperbolic distributions. The resulting formulae were tested in a numerical application to calculate an analytic expression of the expected shortfall of quadratic portfolios with the benefit that moment based sensitivity measures can be derived from the analytic expression. The convergence rate of the analytic expression is fast -one iteration -for small closed integration domains, and slower for open integration domains when compared to the Monte Carlo integration method. The analytic formulae provide a theoretical framework for calculations in robust estimation, robust regression, outlier detection, design of experiments, and stochastic extensions of deterministic elliptical curves results. Keywords: Multivariate truncated moments, Quadratic forms, Elliptical Truncation, Tail moments, Parametric distributions, Elliptical functionsThe first results on truncated moments were concerned with the linear truncated multivariate normal (MVN) distribution, and were provided by Tallis (1961). Tallis (1963) extended the results of linear truncations to the case of elliptical and radial truncation, and Tallis (1965) built on previous results to calculate the moments of a normal distribution with a plane truncation. Mantegna and Stanley (1994) used a truncated Lévy distribution to create a distribution where the sums have slow convergence towards the normal, providing first-and second-order moments. Masoom and Nadarajah (2007) calculated the truncated moments of a generalised Pareto distribution. Arismendi (2013) generalised the results of Tallis (1961) for higher-order moments, and for other elliptical distributions such as the Student's t and the lognormal distributions, and for a finite mixture of multivariate normal distributions. 1In this study, we derived analytical formulae for the calculation of the elliptical truncated moments of the multivariate normal distribution. We calculated an analytical expansion of the elliptical truncated moment generating function (MGF), and then derived this expression for the calculation of the elliptical truncated moments. Previous results on elliptical and radial truncated moments on multivarite normal distributions were provided by Tallis (1963). In this research we used the results of Ruben (1962) to derive the analytical expressions. We then applied the multivariate normal results to derive the multivariate Student's t (MST) and the multivariate generalised hyperbolic (MGH) elliptical truncated moments. Our results can be considered an extension of Ruben's (1962) results for the MST and MGH cases. The importance of elliptical truncated moments' expansions are evident in applications such as the design of experiments (Thompson, 1976;Cameron and Thompson, 1986), robust estimation (Cuesta-Albertos et al., 2008), outlier detection $ The author wish...

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Analytic expressions for integrals of products of spherical Bessel functions

Cited by 49 publications

References 10 publications

Wigner-Eckart theorem in cosmology: Bispectra for total-angular-momentum waves

Wigner-Eckart theorem in cosmology: Bispectra for total-angular-momentum waves

The plane wave expansion, infinite integrals and identities involving spherical Bessel functions

Multivariate elliptical truncated moments

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