# Person: López García, José Luis

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López García

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José Luis

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Matemática e Informática

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0000-0002-6050-9015

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Publication Open Access Formulas for the amplitude of the van der Pol limit cycle through the homotopy analysis method(Hindawi / Wiley, 2009) López García, José Luis; Abbasbandy, S.; López Ruiz, R.; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza; Gobierno de Navarra / Nafarroako Gobernua, Res. 07/05/2008Show more The limit cycle of the van der Pol oscillator, x¨+ε(x2−1)x˙+x=0, is studied in the plane (x,x˙) by applying the homotopy analysis method. A recursive set of formulas that approximate the amplitude and form of this limit cycle for the whole range of the parameter ε is obtained. These formulas generate the amplitude with an error less than 0.1%. To our knowledge, this is the first time where an analytical approximation of the amplitude of the van der Pol limit cycle, with validity from the weakly up to the strongly nonlinear regime, is given.Show more Publication Open Access Generalization of Zernike polynomials for regular portions of circles and ellipses(Optical Society of America, 2014) Navarro, Rafael; López García, José Luis; Díaz, José A.; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika IngeniaritzaShow more Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit circle. Here, we present a generalization of this Zernike basis for a variety of important optical apertures. On the contrary to ad hoc solutions, most of them based on the Gram-Schmidt orthonormalization method, here we apply the diffeomorphism (mapping that has a differentiable inverse mapping) that transforms the unit circle into an angular sector of an elliptical annulus. In this way, other apertures, such as ellipses, rings, angular sectors, etc. are also included as particular cases. This generalization, based on in-plane warping of the basis functions, provides a unique solution and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for most common, elliptical and annular apertures are provided.Show more Publication Open Access Asymptotic behaviour of the Urbanik semigroup(Elsevier, 2015) Berg, Christian; López García, José Luis; Ingeniería Matemática e Informática; Matematika eta Informatika IngeniaritzaShow more We revisit the product convolution semigroup of probability densities ec(t); c > 0 on the positive half-line with moments (n!)c and determine the asymptotic behaviour of ec for large and small t > 0. This shows that (n!)c is indeterminate as Stieltjes moment sequence if and only if c > 2. When c is a natural number ec is a Meijer-G function. From the results about ec we obtain the asymptotic behaviour at 1 of the convolution roots of the Gumbel distribution.Show more Publication 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 MatemáticasShow more 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.Show more Publication 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 PublikoaShow more 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.Show more Publication 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 AMELECShow more 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.Show more Publication 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 PublikoaShow more 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].Show more Publication Open Access A simplification of the stationary phase method: application to the Anger and Weber functions(Kent State University, 2017) López García, José Luis; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e Informática; Universidad Pública de Navarra / Nafarroako Unibertsitate PublikoaShow more The main difficulty in the practical use of the stationary phase method in asymptotic expansions of integrals is originated by a change of variables. The coefficients of the asymptotic expansion are the coefficients of the Taylor expansion of a certain function implicitly defined by that change of variables. In general, this function is not explicitly known, and then the computation of those coefficients is cumbersome. Using the factorization of the exponential factor used in previous works of [Tricomi, 1950], [Erdélyi and Wyman, 1963], and [Dingle, 1973], we obtain a variant of the method that avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler and explicit. On the other hand, the asymptotic sequence is as simple as in the standard stationary phase method: inverse powers of the asymptotic variable. New asymptotic expansions of the Anger and Weber functions Jλx(x) and Eλx(x) for large positive x and real parameter λ 6= 0 are given as an illustration.Show more Publication Open Access New analytic representations of the hypergeometric functions p+1Fp(Springer, 2021) López García, José Luis; Palacios Herrero, Pablo; Pagola Martínez, Pedro Jesús; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y MatemáticasShow more The power series expansions of the hypergeometric functions p+1Fp (a,b1,…,bp;c1,…,cp;z) converge either inside the unit disk |z|<1 or outside this disk |z|>1. Nørlund’s expansion in powers of z/(z−1) converges in the half-plane R(z)<1/2. For arbitrary z0∈C, Bühring’s expansion in inverse powers of z−z0 converges outside the disk |z−z0|= max{|z0|,|z0−1|}. None of them converge on the whole indented closed unit disk |z|≤1,z≠1. In this paper, we derive new expansions in terms of rational functions of z that converge in different regions, bounded or unbounded, of the complex plane that contain the indented closed unit disk. We give either explicit formulas for the coefficients of the expansions or recurrence relations. The key point of the analysis is the use of multi-point Taylor expansions in appropriate integral representations of p+1Fp(a,b1,…,bp;c1,…,cp;z). We show the accuracy of the approximations by means of several numerical experiments.Show more Publication Open Access Convergent expansions of the incomplete gamma functions in terms of elementary functions(World Scientific Publishing, 2017) Bujanda Cirauqui, Blanca; 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 PublikoaShow more We consider the incomplete gamma function γ(a,z) for Ra>0 and z∈C. We derive several convergent expansions of z−aγ(a,z) in terms of exponentials and rational functions of z that hold uniformly in z with Rz bounded from below. These expansions, multiplied by ez, are expansions of ezz−aγ(a,z) uniformly convergent in z with Rz bounded from above. The expansions are accompanied by realistic error bounds.Show more