Abstract We derive an expression for the mutual gravitational force and torque of two bodies having arbitrary shapes and mass distributions, as an expansion in power series of their products of inertia and of the relative coordinates of their centres of mass. The absolute convergence of all the power series developed is rigorously demonstrated. The absence of transcendental functions makes this...

We assess the performance of a discrete-time queueing system with train arrivals. Arrivals at the queue stem from a number of active sessions, each generating a packet in a slot with fixed probability q. Since an exact analysis is not feasible for q 6= 1, we rely on Taylor-series expansions around q = 0 of the joint probability generating functions of the number of active sessions and the queue...

Analytic expressions for the electromagnetic fields of an ultrashort, tightly focused, linearly polarized laser pulse in vacuum are derived from scalar and vector potentials, using a small parameter which assumes a small bandwidth of the laser pulse. The derived fields are compared with those of the Lax series expansion and the complex-source-point approaches and are shown to be well-behaved an...

This paper presents a series expansion that describes the evolution of a mechanical system starting at rest and subject to a time-varying external force. Mechanical systems are presented as second-order systems on a configuration manifold via the notion of affine connections. The series expansion is derived by exploiting the homogeneity property of mechanical systems and the variations of const...

Based on the Hadamard product of power series, polynomial series expansions for confluent hypergeometric functions M(a, c; ·) and for Gaussian hypergeometric functions F (a, b; c; ·) are introduced and studied. It turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions M(a, c; ·) and F (a, b; c; ·), in particular, if the parameters are al...

This article provides a short overview of the theory of First Order Automatic Differentiation (AD) for readers unfamiliar with this topic. In particular, we summarize different characterisations of Forward AD, like the vector-matrix based approach, the idea of lifting functions to the algebra of dual numbers, the method of Taylor series expansion on dual numbers and the application of the push-...

In univariate Padé approximation we learn from the Froissart phenomenon that Padé approximants to perturbed Taylor series expansions of rational functions exhibit almost cancelling polezero combinations that are unwanted. The location of these pole-zero pairs is given by the socalled Froissart polynomial. In this paper the occurrence of the Froissart phenomenon is explored for the first time in...

= 1 1− q1−s , where we have convergence for all s with <(s) > 1. This automatically gives us analytic continuation of ζ to all of C. We note the following simple observations: 1. The Riemann hypothesis: All zeroes of ζ lie on the line <(s) = 1/2, simply because there are no zeroes! 2. ζ has poles at s = 1 + 2πin log q . 3. Locally around s = 1, ζ has the Laurent series expansion: 1 (s− 1) log q...

Based upon the Fourier series expansion, we propose a simple and easy-to-use approach for computing accurate estimates of Black-Scholes double barrier option prices with time-dependent parameters. This new approach is also able to provide tight upper and lower bounds of the exact barrier option prices. Furthermore, this approach can be straightforwardly extended to the valuation of standard Eur...

The initial value problem for the KdV equation @tu+ u@xu + @ 3 xu = 0; u(x; 0) = (x) establishes a nonlinear map K from H(R) to C([ T; T ];H(R)). It has been known for many years that this map K is continuous [2] , [17] and is proved recently being Lipschitz continuous [23]. In this paper it is shown that the nonlinear map K is in nitely many times Frechet di erentiable from H(R) to C([ T; T ];...