David J. Jeffrey

On the LambertW function

The LambertW function is defined to be the multivalued inverse of the functionw →wew. It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containingW.

A sequence of series for the Lambert W function

WC give a uniform treatment of several series expansions for the Lambert Iirfurrction, leading toaninfinite family of new series. Wealsodisrxrss standardization, cornplexbranches,a family of arbitrary-order iterative methods forcornputation of IT”,and give a theorem showing how to correctly solve another simpl~ and frequently occurring nonlinear equation.

The calculation of the low Reynolds number resistance functions for two unequal spheres

The resistance functions that relate the forces, couples, and stresslets exerted on ambient fluid by two unequal rigid spheres in low Reynolds number flow are calculated for the case in which the spheres are immersed in an ambient linear flow. In conjuction with earlier works, this paper completes the tabulation of all of the two‐sphere resistance functions at present needed in investigations of the mechanics of suspensions. Each function is calculated first as a series in inverse powers of the center‐to‐center separation, and then, in order to handle the singular behavior caused by lubrication forces, the asymptotic form which the function takes when the spheres are close is combined with the series expansion into a single expression valid for all separations of the spheres.

The pressure moments for two rigid spheres in low-Reynolds-number flow

The pressure moment of a rigid particle is defined to be the trace of the first moment of the surface stress acting on the particle. A Faxen law for the pressure moment of one spherical particle in a general low‐Reynolds‐number flow is found in terms of the ambient pressure, and the pressure moments of two rigid spheres immersed in a linear ambient flow are calculated using multipole expansions and lubrication theory. The results are expressed in terms of resistance functions, following the practice established in other interaction studies. The osmotic pressure in a dilute colloidal suspension at small Peclet number is then calculated, to second order in particle volume fraction, using these resistance functions. In a second application of the pressure moment, the suspension or particle‐phase pressure, used in two‐phase flow modeling, is calculated using Stokesian dynamics and results for the suspension pressure for a sheared cubic lattice are reported.

The forces and couples acting on two nearly touching spheres in low-Reynolds-number flow

When two unequal spheres are very close, the low-Reynolds-number flow in the narrow gap between them can be analysed using lubrication approximations, and asymptotic formulae for the forces and couples acting on the spheres deduced. The expressions for the forces and couples have previously been regarded as independent, but it is shown here that they are linked by simple physical considerations. The new formulae can be used to improve the accuracy of companion calculations which apply to cases in which the spheres are not close.ZusammenfassungEs wird die Strömung bei kleinen Reynoldszahlen untersucht in einem Spalt zwischen zwei ungleichen Kugeln, die sich nahezu berühren. Asymptotische Formeln für die Kräfte und Momente an den Kugeln werden hergeleitet. Es wird gezeigt daß die Ausdrücke für die Kräfte und Momente, die bis jetzt als unabhängig betrachtet wurden, tatsächlich durch einfache physikalische Betrachtungen verknüpft werden können. Die neuen Formeln können zur Verbesserung der Genauigkeit von Rechnungen für größere Kugelabstände benützt werden.

Mobility functions for two unequal viscous drops in Stokes flow. I. Axisymmetric motions

Analytical results are obtained for mobility functions which describe the hydrodynamic interactions between two unequal viscous drops. It is assumed that the surface tension is sufficiently high so that the drops retain a spherical shape. Exact solutions are introduced for the velocity images for Stokeslets and higher‐order Stokes singularities near a viscous drop and then these image solutions are used to generate expressions valid for all two‐sphere geometries except those for which the gap is much smaller than the diameter of the smaller drop. For rigid spheres, these results are used to obtain a closed‐form expression for the Stokes–Einstein Brownian diffusion coefficient.

The Lambert W function and quantum statistics

We present some applications of the Lambert W function (W function) to the formalism of quantum statistics (QS). We consider the problem of finding extrema in terms of energy for a general QS distribution, which involves the solution of a transcendental equation in terms of the W function. We then present some applications of this formula including Bose–Einstein systems in d dimensions, Maxwell–Boltzmann systems, and black body radiation. We also show that for the appropriate parameter values, this formula reduces to an analytic expression in connection with Wien’s displacement law that was found in a previous study. In addition, we show that for Maxwell–Boltzmann and Bose–Einstein systems, the W function allows us to express the temperature of the system as a function of the thermodynamically relevant chemical potential, the particle density, and other parameters. Finally, we explore an indirect relationship of the W function to the polylogarithm function and to the Lambert transform.

“According to Abramowitz and Stegun” or arccoth needn't be uncouth

This paper addresses the definitions in OpenMath of the elementary functions. The original OpenMath definitions, like most other sources, simply cite [2] as the definition. We show that this is not adequate, and propose precise definitions, and explore the relationships between these definitions.In particular, we introduce the concept of a couth pair of definitions, e.g. of arcsin and arcsinh, and show that the pair arccot and arccoth can be couth.

Approximate polynomial decomposition

where deg g < deg f , deg h < deg f , deg∆f ≤ deg f and ∆f is “small” with respect to the 2-norm of the vector of coefficients. In practice if ‖f‖ denotes the 2-norm of f , then we compute g and h such that ‖∆f‖ is a local minimum with respect to variations in g and h. This problem has been studied for exact polynomials and rational functions by several authors [1, 2, 8, 10, 15, 16]. There are several reasons why approximate polynomial decomposition interests us:

The Turing factorization of a rectangular matrix

The Turing factorization is a generalization of the standard LU factoring of a square matrix. Among other advantages, it allows us to meet demands that arise in a symbolic context. For a rectangular matrix A, the generalized factors are written PA = LDU R, where R is the row-echelon form of A. For matrices with symbolic entries, the LDU R factoring is superior to the standard reduction to row-echelon form, because special case information can be recorded in a natural way. Special interest attaches to the continuity properties of the factors, and it is shown that conditions for discontinuous behaviour can be given using the factor D. We show that this is important, for example, in computing the Moore-Penrose inverse of a matrix containing symbolic entries.We also give a separate generalization of LU factoring to fraction-free Gaussian elimination.