Person:
Ferreira González, Chelo

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https://academica-e.unavarra.es/handle/2454/49026

Last Name

Ferreira González

First Name

Chelo

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Matemática e Informática

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2476

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Now showing 1 - 10 of 21

Open Access
Aprender matemáticas con el ordenador
(Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa, 2004) Bujanda Cirauqui, Blanca; Ferreira González, Chelo; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
Los futuros perfiles profesionales de nuestros actuales alumnos universitarios están cambiando vertiginosamente. Uno de los grandes cambios es el que viene dado por la incorporación del ordenador a la mayoría (casi todos) de estos perfiles. Los profesionales precisan ya un alto nivel de conocimientos de informática, que en el caso más habitual es simplemente nivel de usuario, de manejo del ordenador. Por ello las nuevas tendencias de formación en la universidad deben adaptarse a estas necesidades, e incorporar al aula el ordenador, pero no como un complemento, como se ha venido haciendo hasta ahora con las clases denominadas “de prácticas”, sino como parte esencial del trabajo. Los profesores y los alumnos debemos concienciarnos de que es posible enseñar y aprender con el ordenador. Este es el propósito del libro, un curso de matemáticas básicas, con el ordenador, dirigido a alumnos de primer o primeros cursos de aquellas disciplinas donde las matemáticas no son el eje central pero sí fundamental en su formación (Empresariales, LADE, Ingenierías...). Para ello hemos seleccionado el programa Mathematica, que es el que actualmente utilizamos en la Universidad Pública de Navarra y que nos parece una potente herramienta matemática que además comprende la casi totalidad de las ramas de matemáticas. Por otro lado, este texto puede considerarse también de autoaprendizaje del programa Mathematica, puesto que es un nivel básico y no se necesitan más que los conocimientos elementales de un primer curso de matemáticas. El texto consta de seis capítulos, divididos a su vez en secciones. Los dos primeros introducen el programa y su entorno, el resto describen las opciones básicas y utilidades para un primer acercamiento al cálculo, el álgebra, los gráficos y la estadística descriptiva. Además, al final de cada sección se incluye una serie de ejercicios que recomendamos al alumno, ya que están elegidos de forma que sean un test válido del grado de asimilación de lo abordado en esa sección.
Open Access
Uniform convergent expansions of the Gauss hypergeometric function in terms of elementary functions
(Taylor & Francis, 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
We consider the hypergeometric function 2F1(a, b; c; z) for z ∈ C \ [1,∞). For Ra ≥ 0, we derive a convergent expansion of 2F1(a, b; c; z) in terms of the function (1 − z)−a and of rational functions of z that is uniformly valid for z in any compact in C \ [1,∞). When a ∈ N, the expansion also contains a logarithmic term of the form log(1 − z). For Ra ≤ 0, we derive a convergent expansion of (1 − z)a 2F1(a, b; c; z) in terms of the function (1 − z)−a and of rational functions of z that is uniformly valid for z in any compact in C \ [1,∞) in the exterior of the circle |z − 1| = r for arbitrary r > 0. The expansions are accompanied by realistic error bounds. Some numerical experiments show the accuracy of the approximation.
Open Access
Convergent and asymptotic expansions of solutions of differential equations with a large parameter: Olver cases II and III
(Rocky Mountain Mathematics Consortium, 2015) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
This paper continues the investigation initiated in [Lopez, 2013]. We consider the asymptotic method designed by F. Olver [Olver, 1974] for linear differential equations of the second order containing a large (asymptotic) parameter . We consider here the second and third cases studied by Olver: differential equations with a turning point (second case) or a singular point (third case). It is well-known that his method gives the Poincar´e-type asymptotic expansion of two independent solutions of the equation in inverse powers of . In this paper we add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green function of an auxiliary problem, we transform the initial value problem into a Volterra integral equation of the second kind. Then, using a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence has also the property of being an asymptotic expansion for large (not of Poincar´e-type) of the solution of the problem. Moreover, we show
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
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
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
On a modifcation of Olver's method: a special case
(Springer US, 2016) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
We consider the asymptotic method designed by Olver (Asymptotics and special functions. Academic Press, New York, 1974) for linear differential equations of the second order containing a large (asymptotic) parameter : xm y −2 y = g(x)y, with m ∈ Z and g continuous. Olver studies in detail the cases m = 2, especially the cases m = 0, ±1, giving the Poincaré-type asymptotic expansions of two independent solutions of the equation. The case m = 2 is different, as the behavior of the solutions for large is not of exponential type, but of power type. In this case, Olver’s theory does not give many details. We consider here the special case m = 2. We propose two different techniques to handle the problem: (1) a modification of Olver’s method that replaces the role of the exponential approximations by power approximations, and (2) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter.
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
The asymptotic expansion of the swallowtail integral in the highly oscillatory region
(Elsevier, 2018) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
The mathematical models of many short wavelength phenomena, specially wave propagation and optical diffraction, contain, as a basic ingredient, oscillatory integrals with several nearly coincident stationary phase or saddle points. The uniform approximation of those integrals can be expressed in terms of certain canonical integrals and their derivatives [2,16]. The importance of these canonical diffraction integrals is stressed in [14] by means of the following sentence: The role played by these canonical diffraction integrals in the analysis of caustic wave fields is analogous to that played by complex exponentials in plane wave theory. Apart from their mathematical importance in the uniform asymptotic approximation of oscillatory integrals [12], the canonical diffraction integrals have physical applications in the description of surface gravity waves [11], [17], bifurcation sets, optics, quantum mechanics, chemical physics [4] and acoustics (see [1], Section 36.14 and references there in). To our knowledge, the first application of this family of integrals traces back to the description of the disturbances on a water surface produced, for example, by a traveling ship. These disturbances form a familiar pattern of bow and stern waves which was first explained mathematically by Lord Kelvin [10] using these integrals.