We outline a method for computing the tangent cone of a space curve at any of its points. We rely on the theory of regular chains and Puiseux series expansions. Our approach is novel in that it explicitly constructs the tangent cone at arbitrary and possibly irrational points without using a standard basis.

where 1 2 1 2 , , , r r θ θ are any real numbers, and n is any positive integer. We can obtain the infinite series forms of these double integrals using Taylor series expansions and integration term by term theorem; these are the major results of this paper (i.e., Theorems 1-3). Adams et al. [1], Nyblom [2], and Oster [3] provided some techniques to solve the integral problems. Yu [4-29], Yu an...

In this paper, we present an implementation of the harmonic polylog-arithm of Remiddi and Vermaseren [1] for Mathematica. It contains an implementation of the product algebra, the derivative properties, series expansion and numerical evaluation. The analytic continuation has been treated carefully, allowing the user to keep the control over the definition of the sign of the imaginary parts. Man...

We study positive random variables whose moments can be expressed by products and quotients of Gamma functions; this includes many standard distributions. General results are given on existence, series expansion and asymptotics of density functions. It is shown that the integral of the supremum process of Brownian motion has moments of this type, as well as a related random variable occurring i...

It is well known* that certain types of partial differential equation may be solved using integral transforms with suitable kernels. In general, these equations may be solved by the classical method of separating variables, but the use of an integral transform yields the solution in a more direct way in the sense that the boundary values are contained in the solution. It is the purpose of this ...

In this paper we study the action of a double infinite matrix A on f ∈ H ν (weighted Banach space, 1 ≤ p ≤ ∞) and on its wavelet coefficients. Also, we find the frame condition for A−transform of f ∈ H ν whose wavelet series expansion is known.

We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide tables of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Apéry-like differential equations.

We developed the numerical procedures to evaluate the inverse functions of the complete elliptic integrals of the first and second kind, K(m) and E(m), with respect to the parameter m. The evaluation is executed by inverting eight sets of the truncated Taylor series expansions of the integrals in terms of m or of − log(1 −m). The developed procedures are (1) so precise that the maximum absolute...

The long-range asymptotic expression for the multipole expansion of the retarded interatomic dispersion energy is shown to consist of contributions from electric dipole-dipole, dipolequadrupole and quadrupole-quadrupole interactions, all varying as the inverse seventh power of the interatomic separation. The general expressions for these interactions lead to short-range series expansions which ...

Absfracf-This paper presents a general approach to the derivation of series expansions of second-order wide-sense stationary mean-square continuous random process valid over an infinite-time interval. The coefficients of the expansion are orthogonal and convergence is in the mean-square sense. The method of derivation is based on the integral representation of such processes. It covers both the...