Open Access
Orthogonal basis with a conicoid first mode for shape specification of optical surfaces
(Optical Society of America, 2016) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Navarro, Rafael; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed.
Open Access
Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals
(Wiley, 2023) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We consider the highly oscillatory integral 𝐹(𝑤) ∶= ∫ ∞ −∞ 𝑒𝑖𝑤(𝑡𝐾+2+𝑒𝑖𝜃𝑡𝑝) 𝑔(𝑡)𝑑𝑡 for large positive values of 𝑤, −𝜋 < 𝜃 ≤ 𝜋, 𝐾 and 𝑝 positive integers with 1 ≤ 𝑝 ≤ 𝐾, and 𝑔(𝑡) an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when 𝑤 → +∞ for general values of 𝐾 and 𝑝 in terms of elementary functions, and determine the Stokes lines. For 𝑝 ≠ 1, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters 𝐾 and 𝑝; the special case 𝑝=1 requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals Ψ𝐾(𝑥1, 𝑥2,…,𝑥𝐾) for large values of one of its variables, say 𝑥𝑝, and bounded values of the remaining ones. This family of integrals may be written in the form 𝐹(𝑤) for appropriate values of the parameters 𝑤, 𝜃 and the function 𝑔(𝑡). Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large |𝑥𝑝|. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al.
Open Access
An extension of the multiple Erdélyi-Kober operator and representations of the generalized hypergeometric functions
(De Gruyter, 2018) Karp, D. B.; López García, José Luis; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
In this paper we investigate the extension of the multiple Erd elyi-Kober fractional integral operator of Kiryakova to arbitrary complex values of parameters by the way of regularization. The regularization involves derivatives of the function in question and the integration with respect to a kernel expressed in terms of special case of Meijer's G function. An action of the regularized multiple Erd elyi-Kober operator on some simple kernels leads to decomposition formulas for the generalized hypergeometric functions. In the ultimate section, we de ne an alternative regularization better suited for representing the Bessel type generalized hypergeometric function p􀀀1Fp. A particular case of this regularization is then used to identify some new facts about the positivity and reality of zeros of this function.
Open Access
New recurrence relations for several classical families of polynomials
(Taylor and Francis, 2021) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
In this paper, we derive new recurrence relations for the following families of polynomials: nörlund polynomials, generalized Bernoulli polynomials, generalized Euler polynomials, Bernoulli polynomials of the second kind, Buchholz polynomials, generalized Bessel polynomials and generalized Apostol–Euler polynomials. The recurrence relations are derived from a differential equation of first order and a Cauchy integral representation obtained from the generating function of these polynomials.
Open Access
The Pearcey integral in the highly oscillatory region
(Elsevier, 2016) López García, José Luis; Pagola Martínez, Pedro Jesús; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e Informática; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We consider the Pearcey integral P(x, y) for large values of |y| and bounded values of |x|. The integrand of the Pearcey integral oscillates wildly in this region and the asymptotic saddle point analysis is complicated. Then we consider here the modified saddle point method introduced in [Lopez, Pérez and Pagola, 2009] [4]. With this method, the analysis is simpler and it is possible to derive a complete asymptotic expansion of P(x, y) for large |y|. The asymptotic analysis requires the study of three different regions for separately. In the three regions, the expansion is given in terms of inverse powers of y2/3 and the coefficients are elementary functions of x. The accuracy of the approximation is illustrated with some numerical experiments.
Open Access
Convergent and asymptotic methods for second-order difference equations with a large parameter
(Springer, 2018) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e Informática; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We consider the second-order linear difference equation y(n+2)−2ay(n+1)−Λ2y(n)=g(n)y(n)+f(n)y(n+1) , where Λ is a large complex parameter, a≥0 and g and f are sequences of complex numbers. Two methods are proposed to find the asymptotic behavior for large |Λ|of the solutions of this equation: (i) an iterative method based on a fixed point method and (ii) a discrete version of Olver’s method for second-order linear differential equations. Both methods provide an asymptotic expansion of every solution of this equation. The expansion given by the first method is also convergent and may be applied to nonlinear problems. Bounds for the remainders are also given. We illustrate the accuracy of both methods for the modified Bessel functions and the associated Legendre functions of the first kind.
Open Access
Effect of high-energy ball-milling on the magnetostructural properties of a Ni45Co5Mn35Sn15 alloy
(Elsevier, 2021) López García, José Luis; Sánchez-Alarcos Gómez, Vicente; Recarte Callado, Vicente; Rodríguez Velamazán, José Alberto; Unzueta, Iraultza; García, José Ángel; Plazaola, Fernando; La Roca, Paulo Matías; Pérez de Landazábal Berganzo, José Ignacio; Zientziak; Institute for Advanced Materials and Mathematics - INAMAT2; Ciencias; Gobierno de Navarra / Nafarroako Gobernua, PC017-018 AMELEC
The effect of high-energy ball-milling on the magnetostructural properties of a Ni45Co5Mn35Sn15 alloy in austenitic phase at room temperature has been analyzed by neutron and high-resolution X-ray diffraction. The ball milling promotes a mechanically-induced martensitic transformation as well as the appearance of amorphous-like non-transforming regions, following a double stage; for short milling times (below 30 min), a strong size reduction and martensite induction occur. On the opposite, for longer times, the increase of strains predominates and consequently a larger amount of non-transforming regions appears. The effect of the microstructural defects brought by milling (as dislocations) on both the enthalpy change at the martensitic transformation and the high field magnetization of the austenite has been quantitatively estimated and correlated to the internal strains. Contrary to what occurs in ternary Ni-Mn-Sn alloys, the mechanically-induced defects do not change the ferromagnetic coupling between Mn atoms, but just cause a net reduction on the magnetic moments.
Open Access
Series representations of the Volterra function and the Fransén–Robinson constant
(Elsevier, 2021) López García, José Luis; Pagola Martínez, Pedro Jesús; Palacios Herrero, Pablo; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
The Volterra function μ(t,β,α) was introduced by Vito Volterra in 1916 as the solution to certain integral equations with a logarithmic kernel. Despite the large number of applications of the Volterra function, the only known analytic representations of this function are given in terms of integrals. In this paper we derive several convergent expansion of μ(t,β,α) in terms of incomplete gamma functions. These expansions may be used to implement numerical evaluation techniques for this function. As a particular application, we derive a numerical series representation of the Fransén–Robinson constant F := µ(1, 1, 0) = R ∞ 0 1 Γ(x) dx. Some numerical examples illustrate the accuracy of the approximations
Open Access
Analytic formulas for the evaluation of the Pearcey integral
(American Mathematical Society, 2017) López García, José Luis; Pagola Martínez, Pedro Jesús; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We can find in the literature several convergent and/or asymptotic expansions of the Pearcey integral P(x, y) in different regions of the complex variables x and y, but they do not cover the whole complex x and y planes. The purpose of this paper is to complete this analysis giving new convergent and/or asymptotic expansions that, together with the known ones, cover the evaluation of the Pearcey integral in a large region of the complex x and y planes. The accuracy of the approximations derived in this paper is illustrated with some numerical experiments. Moreover, the expansions derived here are simpler compared with other known expansions, as they are derived from a simple manipulation of the integral definition of P(x, y).
Open Access
Uniform convergent expansions of the error function in terms of elementary functions
(Springer, 2023) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We derive a new analytic representation of the error function erfz in the form of a convergent series whose terms are exponential and rational functions. The expansion holds uniformly in z in the double sector | arg (±z) | <π/4. The expansion is accompanied by realistic error bounds.