Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55)

Abstract

t he spectroscopy of transition-group complexes, Kihara the interact.ions of convex molecules, Koide and Oguchi t he magnetic properties of compounds, Liehr forbidden t.ransitions, :l\IcLennan the formal statist.ical theOl'~' of t.l'Unsport proces es, Scrocco the interpretation of quaClrupole coupling data, and finally Widom the collision t heory of reactiolt rates.This reviewer foulld t he contributions by IGhara and Widom the most st.imulating and interesting. The former is a clear :1ccount of how measured quantities such as the second virial coefficient, the diffusion coefficient, 01' the crystal st.ruc ture of mo lecllles may be interpreted on the basis of a model which takes intermolecula r for ces as acting, not between the molecular centers, but between convex cores assumed to exist inside the molecules. Widom gives n penetrating and exceptionally welI-written account of recent developments in the collision theory of bimolecular reactious. For in. tance, he discusses how the redistribution of molecules into inlel'llal states, such as occur:; in the vibrat.ional relaxation of diatomic molecules, is perturbed when c!issociatioll also takes place.As is customary in books of this type, there is no particular connect.ion between t.he various articles, and they are written in whatever forlll most appeals to the nuthor. The result is eO'ecLively a jou l'llal of U lII'ela ted review articles. Each co ntribu-t·ion should be of interest to so me groups of readers, but few will want to read them all, or even most of them, with more th:1n superficialatlention.

Neutrino emission from neutron stars

“…Substituting the expression for f ′ e in Eq. (24) and taking the factor exp(−µ e /T ) out of the integration, we find that the remaining integral depends only on one parameter t r = k B T /(m e c 2 ) = 0.1686 T 9 , where T 9 = T /10 9 K, and can be expressed through a McDonald function (e.g., Abramowitz and Stegun 1964). In the limiting cases t r ≪ 1 (nonrelativistic positrons) and t r ≫ 1 (ultrarelativistic positrons) the integral is done analytically.…”

Efficient and robust computation of PDF features from diffusion MR signal

“…where ζ denotes the scale factor, L (α) n (x) are the generalized Laguerre polynomials which are the eigenfunctions of the FourierBessel transform [54]. The notation (a) n denotes the Pochhammer symbol and 1F1 denotes the confluent hyper-geometric function.…”

VLF waves at satellite altitude to investigate Earth electrical conductivity

“…For orders n larger than the Bessel function argument, it is necessary to compute the ratios of the Bessel functions and of their derivative using recursion relationships obtained from the recursion equations [Abramowitz and Stegun, 1964]. The use of quadruple precision can be an alternative solution but is faced to enormous memory requirement and long computation time without preventing numerical instabilities.…”

“…Owing to the behavior of the Bessel functions with respect to their order and argument, they are calculated differently for low and high orders. For the low orders n the Bessel function are calculated using the recursion equations [Abramowitz and Stegun, 1964]. When the order n is close to the Bessel function argument for large values, the numerical calculation of the Bessel functions is an issue and widely discussed [Berry, 1964;Houdzoumis, 1994] and is still the most challenging part in this study.…”