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Full-Text Articles in Physical Sciences and Mathematics
The Fourier–Legendre Series Of Bessel Functions Of The First Kind And The Summed Series Involving 1F2 Hypergeometric Functions That Arise From Them, Jack C. Straton
The Fourier–Legendre Series Of Bessel Functions Of The First Kind And The Summed Series Involving 1F2 Hypergeometric Functions That Arise From Them, Jack C. Straton
Physics Faculty Publications and Presentations
The Bessel function of the first kind JN(kx) is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind IN(kx). The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for JN(kx) useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in …
Integral Representations Over Finite Limits For Quantum Amplitudes, Jack C. Straton
Integral Representations Over Finite Limits For Quantum Amplitudes, Jack C. Straton
Physics Faculty Publications and Presentations
We extend previous research to derive three additional M-1-dimensional integral representations over the interval [0,1]" The prior version covered the interval [0,∞]" role="presentation position: relative;">[0,∞][0,∞]. This extension applies to products of M Slater orbitals, since they (and wave functions derived from them) appear in quantum transition amplitudes. It enables the magnitudes of coordinate vector differences (square roots of polynomials) |x1−x2|=x12−2x1x2cosθ+x22" to be shifted from disjoint products of functions into a single quadratic form, allowing for the completion of its square. The M-1-dimensional integral representations of M Slater orbitals that both this extension and the prior version introduce …
An Integral Transform For Quantum Amplitudes, Jack C. Straton
An Integral Transform For Quantum Amplitudes, Jack C. Straton
Physics Faculty Publications and Presentations
The central impediment to reducing multidimensional integrals of transition amplitudes to analytic form, or at least to a fewer number of integral dimensions, is the presence of magnitudes of coordinate vector differences (square roots of polynomials) |x1−x2|2=x21−2x1x2cosθ+x2 √ in disjoint products of functions. Fourier transforms circumvent this by introducing a three-dimensional momentum integral for each of those products, followed in many cases by another set of integral transforms to move all of the resulting denominators into a single quadratic form in one denominator whose square my be completed. Gaussian transforms introduce a one-dimensional integral for each such product while squaring …