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
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
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
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 II
(Elsevier, 2025-08-01) 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
We consider the Pearcey integral P(x, y) for large values of |x| and bounded values of |y|. The standard saddle point analysis is difficult to apply because the Pearcey integral is highly oscillating in this region. To overcome this problem we use the modified saddle point method introduced in López et al. (2009). A complete asymptotic analysis is possible with this method, and we derive a complete asymptotic expansion of P(x, y) for large |x|, accompanied by the exact location of the Stokes lines. There are two Stokes lines that divide the complex x−plane in two different sectors in which P(x, y) behaves differently when |x| is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of x and y. Both of them are of Poincaré type; one of them is given in terms of inverse powers of x; the other one in terms of inverse powers of x 1/2 , and it is multiplied by an exponential factor that behaves differently in the two mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.
Open Access
Uniform convergent expansions of integral transforms
(American Mathematical Society, 2021) López García, José Luis; Palacios Herrero, Pablo; Pagola Martínez, Pedro Jesús; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Several convergent expansions are available for most of the special functions of the mathematical physics, as well as some asymptotic expansions [NIST Handbook of Mathematical Functions, 2010]. Usually, both type of expansions are given in terms of elementary functions; the convergent expansions provide a good approximation for small values of a certain variable, whereas the asymptotic expansions provide a good approximation for large values of that variable. Also, quite often, those expansions are not uniform: the convergent expansions fail for large values of the variable and the asymptotic expansions fail for small values. In recent papers [Bujanda & all, 2018-2019] we have designed new expansions of certain special functions, given in terms of elementary functions, that are uniform in certain variables, providing good approximations of those special functions in large regions of the variables, in particular for large and small values of the variables. The technique used in [Bujanda & all, 2018-2019] is based on a suitable integral representation of the special function. In this paper we face the problem of designing a general theory of uniform approximations of special functions based on their integral representations. Then, we consider the following integral transform of a function g(t) with kernel h(t, z), F(z) := 1 0 h(t, z)g(t)dt. We require for the function h(t, z) to be uniformly bounded for z ∈D⊂ C by a function H(t) integrable in t ∈ [0, 1], and for the function g(t) to be analytic in an open region Ω that contains the open interval (0, 1). Then, we derive expansions of F(z) in terms of the moments of the function h, M[h(·, z), n] := 1 0 h(t, z)tndt, that are uniformly convergent for z ∈ D. The convergence of the expansion is of exponential order O(a−n), a > 1, when [0, 1] ∈ Ω and of power order O(n−b), b > 0, when [0, 1] ∈/ Ω. Most of the special functions F(z) having an integral representation may be cast in this form, possibly after an appropriate change of the integration variable. Then, special interest has the case when the moments M[h(·, z), n] are elementary functions of z, because in that case the uniformly convergent expansion derived for F(z) is given in terms of elementary functions. We illustrate the theory with several examples of special functions different from those considered in [Bujanda & all, 2018-2019].
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.
Open Access
An analytic representation of the second symmetric standard elliptic integral in terms of elementary functions
(Springer, 2022) Bujanda Cirauqui, Blanca; 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
We derive new convergent expansions of the symmetric standard elliptic integral RD(x,y,z), for x,y,z∈C∖(−∞,0], in terms of elementary functions. The expansions hold uniformly for large and small values of one of the three variables x, y or z (with the other two fixed). We proceed by considering a more general parametric integral from which RD(x,y,z) is a particular case. It turns out that this parametric integral is an integral representation of the Appell function F1(a;b,c;a+1;x,y). Therefore, as a byproduct, we deduce convergent expansions of F1(a;b,c;a+1;x,y). We also compute error bounds at any order of the approximation. Some numerical examples show the accuracy of the expansions and their uniform features.
Open Access
Analysis of singular one-dimensional linear boundary value problems using two-point Taylor expansions
(University of Szeged (Hungría), 2020) 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
We consider the second-order linear differential equation (x2 − 1)y'' + f (x)y′ + g(x)y = h(x) in the interval (−1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet–Neumann). The functions f, g and h are analytic in a Cassini disk Dr with foci at x = ±1 containing the interval [−1, 1]. Then, the two end points of the interval may be regular singular points of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in Dr of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor appro-ximation of the analytic solutions when they exist.
Open Access
The swallowtail integral in the highly oscillatory region II
(Kent State University, 2020) 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
We analyze the asymptotic behavior of the swallowtail integral R ∞ −∞ e i(t 5+xt3+yt2+zt)dt for large values of |y| and bounded values of |x| and |z|. We use the simpli ed saddle point method introduced in [López et al., 2009]. With this method, the analysis is more straightforward than with the standard saddle point method and it is possible to derive complete asymptotic expansions of the integral for large |y| and xed x and z. There are four Stokes lines in the sector (−π, π] that divide the complex y−plane in four sectors in which the swallowtail integral behaves di erently when |y| is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of x, y and z. One of them is of Poincaré type and is given in terms of inverse powers of y 1/2 . The other one is given in terms of an asymptotic sequence of the order O(y −n/9 ) when |y| → ∞, and it is multiplied by an exponential factor that behaves di erently in the four mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.