A Quadruple Integral Involving Product of the Struve Hv(βt) and Parabolic Cylinder Du(αx) Functions

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.

Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation

The impulse response of the fractional oscillation equation was investigated, where the damping term was characterized by means of the Riemann–Liouville fractional derivative with the order α satisfying 0≤α≤2. Two different analytical forms of the response were obtained by using the two different methods of inverse Laplace transform. The first analytical form is a series composed of positive powers of t, which converges rapidly for a small t. The second form is a sum of a damped harmonic oscillation with negative exponential amplitude and a decayed function in the form of an infinite integral, where the infinite integral converges rapidly for a large t. Furthermore, the Gauss–Laguerre quadrature formula was used for numerical calculation of the infinite integral to generate an analytical approximation to the response. The asymptotic behaviours for a small t and large t were obtained from the two forms of response. The second form provides more details for the response and is applicable for a larger range of t. The results include that of the integer-order cases, α= 0, 1 and 2.

An exact integral-to-sum relation for products of Bessel functions

A useful identity relating the infinite sum of two Bessel functions to their infinite integral was discovered in Dominici et al. (Dominici et al. 2012 Proc. R. Soc. A 468 , 2667–2681). Here, we extend this result to products of N Bessel functions, and show it can be straightforwardly proven using the Abel-Plana theorem, or the Poisson summation formula. For N = 2, the proof is much simpler than that of Dominici et al. and significantly enlarges the range of validity.

Infinite Integral Involving the Generalized Modified I-Function of Two Variables

In the present paper, we evaluate the general infinite integral involving the generalized modified I-functions of two variables. At the end, we shall see several corollaries and remarks.

Generation of Surface Waves Due to Initial Axisymmetric Surface Disturbance in Water with a Porous Bottom

A two-dimensional Cauchy Poisson problem for water with a porous bottom generated by an axisymmetric initial surface disturbance is investigated here. The problem is formulated as an initial value problem for the velocity potential describing the motion in the fluid. The Laplace and Hankel transform techniques have been used in the mathematical analysis to obtain the form of the free surface in terms of a multiple infinite integral. This integral is then evaluated asymptotically by the method of stationary phase. The asymptotic form of the free surface is depicted graphically in a number of figures for different values of the porosity parameter and for different types of initial disturbances.

Lineability within Peano Curves, Martingales, and Integral Theory

This paper is devoted to give several improvements of some known facts in lineability approach. In particular, we prove that (i) the set of continuous mappings from the unit interval onto the unit square contains a closed,c-semigroupable convex subset, (ii) the set of pointwise convergent martingales(Xn)n∈NwithEXn→∞isc-lineable, (iii) the set of martingales converging in measure but not almost surely isc-lineable, (iv) the set of sequences(Xn)n∈Nof independent random variables, withEXn=0,∑n=1∞var Xn=∞, and the property that(X1+⋯+Xn)n∈Nis almost surely convergent, isc-lineable, (v) the set of bounded functionsf:[0,1]×[0,1]→Rfor which the assertion of Fubini’s Theorem does not hold is consistent withZFC 1-lineable (it is not 2-lineable), (vi) the set of unbounded functionsf:[0,1]×[0,1]→Rfor which the assertion of Fubini’s Theorem does not hold (with infinite integral allowed) isc-lineable but notc+-lineable.

Fast and high-precision calculation of earth return mutual impedance between conductors over a multilayered soil

Purpose Solutions for the earth return mutual impedance play an important role in analyzing couplings of multi-conductor systems. Generally, the mutual impedance is approximated by Pollaczek integrals. The purpose of this paper is devising fast algorithms for calculation of this kind of improper integrals and its applications. Design/methodology/approach According to singular points, the Pollaczek integral is divided into two parts: the finite integral and the infinite integral. The finite part is computed by combining an efficient Levin method, which is implemented with a Chebyshev differential matrix. By transforming the integration path, the tail integral is calculated with help of a transformed Clenshaw–Curtis quadrature rule. Findings Numerical tests show that this new method is robust to high oscillation and nearly singularities. Thus, it is suitable for evaluating Pollaczek integrals. Furthermore, compared with existing method, the presented algorithm gives high-order approaches for the earth return mutual impedance between conductors over a multilayered soil with wide ranges of parameters. Originality/value An efficient truncation strategy is proposed to accelerate numerical calculation of Pollaczek integral. Compared with existing algorithms, this method is easier to be applied to computation of similar improper integrals, such as Sommerfeld integral.