A unified presentation of the Voigt functions

This paper aims at presenting a unified study of the Voigt functionsK(x,y) andL(x,y) which play a rather important role in several diverse fields of physics such as astrophysical spectroscopy and the theory of neutron reactions. Explicit expressions for these functions are given in terms of relatively more familiar special functions of one and two variables; indeed, each of these representations will naturally lead to various other needed properties of the Voigt functions.     

Spectral line profiles and neutron cross sections: New results concerning the analysis of Voigt functions

The Voigt functions, so important in spectroscopy and neutron physics, are represented as generalized hypergeometric functions (G-functions) of two real variables. A system of partial differential equations for the Voigt functions is derived. By applying Hölder's inequality to an integral representation of the Voigt functions apparently not known in the literature until now, lower and upper bounds are obtained. Moreover, from this representation an asymptotic expansion of Voigt functions for large values of one variable is extracted.     

A unified presentation of the Voigt functions

In the present paper, we have given a generalization of a unified study of the Voigt functionsK(x, y) andL(x, y) obtained by Srivastava and Miller (1987; Vol. 135, pp. 111–118) which play an important role in several diverse fields of physics-such as astrophysical spectroscopy and the theory of neutron reactions. Explicit expressions for these functions are given in terms of relatively more familiar special functions of one and two variables; indeed, each of these representations will naturally lead to various other needed properties of the Voigt functions.     

Some unified presentations of the Voigt functions

The principal object of this note is to provide a natural further step toward the unified presentations of the Voigh functionsK(x, y) andL(x, y) which play a rather important role in such diverse fields of physics as astrophysical spectroscopy and the theory of neutron reactions. Explicit representations for these functions, given in terms of some relatively more familiar special functions of one and two variables, are potentially useful in finding many other needed (numerical or analytical) properties of the Voigt functions. Several erroneous recent contributions to the theory of Voigt functions, including (for example) the main result of A. Siddiqui (1990), are also corrected here.     

Compatibility conditions for a non-quadratic integral of motion

Conditions are found which are satisfied by the coefficients of the expression being a second integral of the motion of an autonomous dynamical system with two degrees of freedom. The coefficientsA, B. , ,E are differentiable functions of the cartesian position coordinatesx, y. The velocity components are denoted by . It is shown that must be constant andB must be of the formB =f(x+y) +g(x-y) wheref, g are arbitrary.Given andB one can always find the remaining coefficientsA, E and also the corresponding potential and second integral. Depending on the specifica case at hand a certain number of arbitrary constants (or arbitrary functions) enter into the potential and the second integral. To each potential (which may be of the separable or nonseparable type in the coordinatesx andy)there corresponds one integral of the above form.     

The angle-dependent redistribution functionsR III andR IV: Line absorption probability distribution functions and reemission coefficients

The line absorption probability distribution functions and the reemission coefficients are derived for the non-coherent scattering functionsR _III andR _IV. The appropriate line profile function forR _III is shown to be a simple Voigt function, while forR _IV, the line absorption probability distribution function is more complex involving a linear combination of two Voigt functions and another more complex probability distribution. The structure of the reemission coefficients forR _III andR _IV is then discussed.     

The representation of the force functions of the Earth's and Moon's gravitation in ellipsoidal coordinates

This paper derives asymptotic expansions of ellipsoidal coordinates in Cartesian coordinates and an expansion in spherical harmonics of the dominant term for the solution of Laplace's equation corresponding to the gravitational force function for a two-dimensional finite body.On comparing the expansion of the dominant term derived here with known expansions of the force functions of the Earth's and Moon's gravitation the author obtains values for the semimajor axes and eccentricities of the singular ellipses of these bodies in terms of the second degree harmonic coefficientsc ₂₀ andc ₂₂.     

Fourier techniques for an analysis of eclipsing binary light curves. VIa

The aim of the present paper is to deduce relations between the integral transformsA _2m, B_2m,andF _1,2 of the light curves of eclipsing binary systems. The integral transformsA _2m, B_2m,andF _1,2 have been related to one another by means of finite or rapidly converging infinite summations obtained by integrations of the series expansions of trigonometric functions.     

A unified theory of polynomial expansions and their applications involving Clebsch-Gordan type linearization relations and Neumann series

Motivated by their potential for applications in several diverse fields of physical, astrophysical, and engineering sciences, this paper aims at presenting a unified study of various classes of polynomial expansions and multiplication theorems associated with the general multivariable hypergeometric function (studied recently by A. W. Niukkanen and H. M. Srivastava), which provides an interesting and useful unifiation of numerous families of special functions in one and more variables, encoutered naturally (and rather frequently) in many physical, quantum chemical, and quantum mechanical situations. Several interesting applications of these general polynomial expansions are considered, not only in the derivations of various Clebsch-Gordan type linearization relations involving products of several Jacobi or Laguerre polynomials, but also to associated Neumann expansions in series of the Bessel functionsJ _v (z) andI _v (z) (and of their suitable products).     

Solution of the three-dimensional inverse problem

The three dimensional inverse problem for a material point of unit mass, moving in an autonomous conservative field, is solved. Given a two-parametric family of space curvesf(x, y, z)=c ₁,g(x, y, z)=c ₂, it is shown that, in general, no potentialU=U(x, y, z) exists which can give rise to this family. However, if the given functionsf(x, y, z) andg(x, y, z) satisfy certain conditions, the corresponding potentialU(x, y, z), as well as the total energyE=E(f, g) are determined uniquely, apart from a multiplicative and an additive constant.     

Asymptotic theory of the transfer of polarized radiation with resonance scattering in the doppler core of the line

In the first of the series of papers by Ivanov et al. it was shown that the model problem of the transfer of polarized radiation as a result of resonance scattering from two-level atoms in a homogeneous plane atmosphere in the absence of LTE comes down, in the approximation of complete frequency redistribution, to the solution of an integral matrix equation of the Wiener-Hopf type for a (2 × 2) matrix source function S(τ). In the second paper in this series, devoted to the vector Milne problem, complete asymptotic expansions of the matrix I(z) [which is essentially a Laplace transform of the matrix S(τ)] for the case of a Doppler profile of the coefficient of absorption, and the coefficients of asymptotic expansions of S(τ) (τ » 1) are expressed in terms of coefficients of the expansions of I(z). We show that asymptotic expansions of S(τ) can be found directly from an integral matrix equation of the Wiener-Hopf type for S(τ). We give new recursive equations for the coefficients of these expansions, as well as a new derivation of asymptotic expansions of the matrix I, including its second column, which was considered only briefly by Ivanov et al.     

Boundary curves for families of planar orbits

For monoparametric familiesf(x,y)=c of planar orbits, created by a planar potentialV(x,y), we introduce the notion of the family boundary curves (FBC). All members of the familyf(x,y)=c are traced in an allowable region of thexy plane, defined by the corresponding FBC, with total energyE=E(c) varying along the family. Family boundary curves are also found for two-parametric familiesf(x,y,b)=c. The relation of equilibrium points and asymptotic orbits, possibly possessed by the potentialV(x,y), to be FBC is studied.     

Recurrence formulae for the Hansen's developments

In this paper we derive some recurrence formulae which can be used to calculate the Fourier expansions of the functions (r/a) ⁿ cosmv and (r/a) ⁿ sinmv in terms of the eccentric anomalyE or the mean anomalyM. We also establish a recurrence process for computing the series expansions for alln andm when the expansions of two basic series are known. These basic series were given in explicit form in the classical literature. The recurrence formulae are linear in the functions involved and thus make very simple the computation of the series.This work was supported by NASA contract No. NASr 54(06).—The paper was presented at the AIAA/AAS meeting, Princeton University, August 1969.     

Line formation in moving media: Asymptotic expansions of some special functions

Line formation in the spectrum of a moving medium with a spherical geometry is considered. In the Sobolev approximation there are some special functions that determine the source function and the force of radiation pressure in the line. The most important case is that of a small dimensionless velocity gradient (i.e., a large dimensionless Sobolev length τ) and a small ratio β of the opacity in the continuum to the opacity in the line. Until now there has been no detailed analytical information about the asymptotic behavior of these functions. For the case of a Doppler profile of the absorption coefficient, we clarify the nontrivial structure of their total asymptotic expansions for τ » 1, β « 1, and arbitrary Βτ. We give an algorithm for obtaining all the coefficients of these expansions and give explicit expressions for the first few coefficients. We also compare the asymptotic expansions with the numerical calculations of these functions available in the literature. We also briefly consider the case of a power-law decrease in the absorption coefficient in the line wing (and, in more detail, the case of Lorentz wings of the Voigt profile).     

Intrinsic stability of periodic orbits

Families of orbits of a conservative, two degree-of-freedom system are represented by an unsteady velocity field with componentsu(x, y, t) andv(x, y, t). Intrinsic stability properties depend on velocity field divergence and curl, whose dynamical evolution is determined by a matrix Riccati equation. Near equilibrium, divergence-free or irrotational fields are dynamically compatible with the conservative force field. It is shown that a necessary condition for stable periodic orbits is satisfied when the orbitaveraged divergence is zero, which results in bounded normal variations. A sufficient condition for stability is derived from the requirement that tangential variations do not exhibit secular growth.In a steady, divergence-free field, velocity component functionsu(x, y) andv(x, y) may be continuedanalytically from any initial condition, except when velocity is parallel to U or at equilibria. In an unsteady field, the orbit-averaged divergence is zero when the vorticity function is periodic. When such a field exists, initial conditions for stable periodic orbits (i.e., characteristic loci) may be determinedanalytically.     

Restricted quantum-mechanical three-body problems

Eigenvalues and normalized wave functions of an electron are derived in the field of a generalized dipole with chargesZ ₁ andZ ₂ (Z ₁+Z ₂ 0) in the asymptotic region up to the third order (inr ₂) where the distance,r ₂, between the two charges is small. These asymptotic wave functions render it possible to calculate the asymptotic expansion for the coefficients of a coupled infinite system of second order differential equations arising from a perturbative (analytic) solution to the Schrödinger equation of helium-like ions if the nuclear charge is not less than 2.     

Fourier analysis of the light curves of eclipsing variables. XIII

New expressions for the fractional loss of light _l ⁰ have been derived in the simple forms of rapidly converging expansions to the series of Chebyshev polynomials, Jacobi polynomials, and Kopal'sJ-integrals. In these expansions, which are a supplement to those given by Kopal (1977b), variablesk andh occur in different products that simplify the numerical computation. The treatment follows the new definition of _l ⁰ which has been recently developed by Kopal (1977a).     

Inverse problem with two-parametric families of planar orbits

As a possible extension of recent work we study the following version of the inverse problem in dynamics: Given a two-parametric familyf(x, y, b)=c of plane curves, find an autonomous dynamical system for which these curves are orbits.We derive a new linear partial differential equation of the first order for the force componentsX(x, y) andY(x, y) corresponding to the given family. With the aid of this equation we find that, depending on the given functionf, the problem may or may not have a solution. Based on given criteria, we present a full classification of the various cases which may arise.     

The solution of radiation transfer problem for a slab with space-dependent albedo and anisotropic scattering

The transport of thermal radiation has been considered within a finite slab which absorb and scatter anisotropically. The problem involves the space-dependent single-scattering albedow(x). Two approximations are taken forw(x). In the first it is represented in exponential form asw(x)=w ₀ exp(–x/s), wherew ₀ ands are given constants andx is the optical variable. The second approximation assumes the formw(x) = _r=0 ^R d _r ^* p _r (x/a), whered _r ^* are known expansion coefficients anda is the half optical thickness of the slab. Analytic expressions for the forward, backward radiation intensities and fluxes are given in each approximation. The solution of the linear transport equation is performed on the basis of integral Fourier transforms.     

The influence of a resisting medium on 3-dimensional periodic orbits of the restricted 3-body problem

Summary The evolution of the extreme values of the functionsx(t),y(t) andz(t) which are the coordinates of the third bodyP in the barycentric rotating frame of reference, when friction is present, is discussed. These values, which are constant for periodic orbits, change due to the presence of the resisting medium. It is shown that either the orbit tends to become circular and coplanar with the two primaries or to collide with one of the primaries.