Formulas and Theorems for the Special Functions of Mathematical Physics

Magnus, Wilhelm; Oberhettinger, Fritz; Soni, Raj Pal

doi:10.1007/978-3-662-11761-3

Cited by 2,508 publications

(817 citation statements)

References 1 publication

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“…The values for the root mean square of ϕ(φ) compare quite favourably, in fact, with the heuristic [20]) to be at most 2 √ π /8. The upshot, of course, is that in all our configurations (except possibly the ' o' ones), it is fair to say that equidistribution is seen to be taking hold in a generally robust manner along the various arcs γ .…”

Section: Instead Of I/s(0)s(t)supporting

confidence: 53%

On correlations of CM-type Maass waveforms under the horocyclic flow

Hejhal¹

2003

J. Phys. A: Math. Gen.

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Maass waveforms of CM-type are a special kind of eigenfunction of the hyperbolic Laplacian whose 'defining' components (namely eigenvalue and Fourier coefficients) are given by simple formulae involving algebraic integers chosen from a suitable number field K/Q. In this paper, we report on some computer experiments aimed at ascertaining the extent to which the autocorrelation behaviour of CM-forms agrees with that of 'mock' (i.e. random) waveforms in the limit of high energy. Our results suggest that no significant differences are seen.PACS number: 05.45.Mt BackgroundQuantum chaos is a topic of natural interest on negatively curved Riemannian manifolds M. This is a reflection of the well-known fact that the dynamics of a classical point particle traversing M are typically ergodic (cf e.g. [10][11][12]23] and the literature cited there). The simplest class of such manifolds is that associated with the set-up where M is expressed as a quotient space \H , being a discrete subgroup of SL(2, R) and H the Poincaré upper half-plane {Im(z) > 0}. To ensure that \H has good geometry in the hyperbolic metric ds 2 = dx 2 + dy 2 y 2 (the curvature here being −1), one typically assumes that the hyperbolic areaof M is finite. From a dynamical perspective, a good portion of quantum chaos is then concerned with determining the extent to which high-frequency eigenfunctions of the hyperbolic Laplacian on \H simulate random waves as the frequency (or wave number) tends to infinity.

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Section: Instead Of I/s(0)s(t)supporting

confidence: 53%

On correlations of CM-type Maass waveforms under the horocyclic flow

Hejhal¹

2003

J. Phys. A: Math. Gen.

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“…where J m is a Bessel function and we have used a known [11,13] integral expression for J m . The expressions given by eqn.…”

Section: The Su(2) → E(2) Contraction Limitmentioning

confidence: 99%

Asymptotic limits of SU(2) and SU(3) Wigner functions

Rowe

Guise

Sanders

2001

Journal of Mathematical Physics

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“…[6]) and ψ n (see e.g. [18]). However, this does not fit to our convention on the utilization of capital letters for scalar functions.…”

Section: Theorem 44 the Separation Ansatzmentioning

confidence: 99%

“…is not necessary since the corresponding expression in the integrand of (18) can be reduced to the multiplication of a vector with a euclidean scalar product: …”

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confidence: 99%

Theoretical aspects of a multiscale analysis of the eigenoscillations of the earth

Michel¹

2003

Rev. Mat. Complut.

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The elastic behaviour of the Earth, including its eigenoscillations, is usually described by the Cauchy-Navier equation. Using a standard approach in seismology we apply the Helmholtz decomposition theorem to transform the Fourier transformed Cauchy-Navier equation into two non-coupled Helmholtz equations and then derive sequences of fundamental solutions for this pair of equations using the Mie representation. Those solutions are denoted by the Hansen vectors Ln,j, Mn,j, and Nn,j in geophysics. Next we apply the inverse Fourier transform to obtain a function system depending on time and space. Using this basis for the space of eigenoscillations we construct scaling functions and wavelets to obtain a multiresolution for the solution space of the Cauchy-Navier equation.2000 Mathematics Subject Classification: 35J05, 42C40.

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Formulas and Theorems for the Special Functions of Mathematical Physics

Cited by 2,508 publications

References 1 publication

On correlations of CM-type Maass waveforms under the horocyclic flow

On correlations of CM-type Maass waveforms under the horocyclic flow

Asymptotic limits of SU(2) and SU(3) Wigner functions

Theoretical aspects of a multiscale analysis of the eigenoscillations of the earth

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