The Hilbert transform has many uses, including solving problems in aerodynamics, condensed matter physics, optics, fluids, and engineering. Written in a style that will suit a wide audience (including the physical sciences), this book will become the reference of choice on the topic, whatever the subject background of the reader. It explains all the common Hilbert transforms, mathematical techniques for evaluating them, and has detailed discussions of their application. Especially useful for researchers are the tabulation of analytically evaluated Hilbert transforms, and an atlas that immediately illustrates how the Hilbert transform alters a function. A collection of exercises helps the reader to test their understanding of the material in each chapter. The bibliography is a wide-ranging collection of references both to the classical mathematical papers, and to a diverse array of applications.

Surface roughness and the enhanced intensity of Raman scattering by molecules adsorbed on metals

Extensive variational calculations using Hylleraas-type functions are carried out on the S ground states for the following members of the Li I isoelectronic series: Li I, B III, C Iv, N v, 0 vI, F vII, and NevIII. For each species, the nonrelativistic energy, the electronic density at the nucleus, the specific mass shift, the transition isotope shift, the Fermi contact interaction, the diamagnetic susceptibility, the nuclear magnetic shielding factor, moments of (r;") for n = -1, 1, 2, and 3 and of (r;, ") for n = -1, 1, and 2 are reported. The nonrelativistic energies obtained are lower than any previously published values for each member of the Li I series examined. The effect of one versus two spin eigenfunctions in the basis set is examined. The rates of convergence for both the oneand two-spin-eigenfunction basis sets are examined for each expectation value.

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Convergence accelerator methods are employed to analyze some of the most difficult three-electron integrals that arise in atomic calculations. These integrals have an explicit dependence on the interelectronic coordinates, and take the form integral r(i)(1)r(j)(2)r(k)(3)r(l)(23)r(m)(31)r(n)(12) exp((-alpha(r1)-beta(r2)-gamma(r3))dr(1)dr(2)dr(3). The focus of the present investigation are the most difficult cases of the parameter set [i, j, k, l, m, n]. Several convergence accelerator techniques are studied, and a comparison presenting the relative effectiveness of each technique is reported. When the convergence accelerator approach is combined with specialized numerical quadrature methods, we find that the overall technique yields high-precision results and is fairly efficient in terms of computational resources. Difficulties associated with the standard numerical precision loss of convergence accelerator techniques are circumvented.