Open Access
Orthogonal basis for the optical transfer function
(Optical Society of America, 2016) Ferreira González, Chelo; López García, José Luis; Navarro, Rafael; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
We propose systems of orthogonal functions qn to represent optical transfer functions (OTF) characterized by including the diffraction-limited OTF as the first basis function q0 OTF perfect. To this end, we apply a powerful and rigorous theoretical framework based on applying the appropriate change of variables to well-known orthogonal systems. Here we depart from Legendre polynomials for the particular case of rotationally symmetric OTF and from spherical harmonics for the general case. Numerical experiments with different examples show that the number of terms necessary to obtain an accurate linear expansion of the OTF mainly depends on the image quality. In the rotationally symmetric case we obtained a reasonable accuracy with approximately 10 basis functions, but in general, for cases of poor image quality, the number of basis functions may increase and hence affect the efficiency of the method. Other potential applications, such as new image quality metrics are also discussed.
Open Access
New series expansions of the 3F2 function
(2015) López García, José Luis; Pagola Martínez, Pedro Jesús; Pérez Sinusía, Ester; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
We can use the power series definition of 3F2(a1, a2, a3; b1, b2; z) to compute this function for z in the unit disk only. In this paper we obtain new expansions of this function that are convergent in larger domains. Some of these expansions involve the polynomial 3F2(a1,−n, a3; b1, b2; z) evaluated at certain points z. Other expansions involve the Gauss hypergeometric function 2F1. The domain of convergence is sometimes a disk, other times a half-plane, other times the region |z|2 < 4|1 − z|. The accuracy of the approximation given by these expansions is illustrated with numerical experiments.
Open Access
The use of two-point Taylor expansions in singular one-dimensional boundary value problems I
(Elsevier, 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 second-order linear differential equation (x + 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(x), g(x) and h(x) are analytic in a Cassini disk Dr with foci at x = ±1 containing the interval [−1, 1]. Then, the end point of the interval x = −1 may be a regular singular point 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 approximation of the analytic solutions when they exist.
Open Access
Uniformly convergent expansions for the generalized hypergeometric functions p –1Fp and pFp
(Taylor & Francis, 2020) López García, José Luis; Pagola Martínez, Pedro Jesús; Karp, D. B.; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
We derive a convergent expansion of the generalized hypergeometric function p−1 F p in terms of the Bessel functions 0 F 1 that holds uniformly with respect to the argument in any horizontal strip of the complex plane. We further obtain a convergent expansion of the generalized hypergeometric function p F p in terms of the confluent hypergeometric functions 1 F 1 that holds uniformly in any right half-plane. For both functions, we make a further step forward and give convergent expansions in terms of trigonometric, exponential and rational functions that hold uniformly in the same domains. For all four expansions we present explicit error bounds. The accuracy of the approximations is illustrated by some numerical experiments.
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
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
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 Publikoa
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.
Embargo
Convergent and asymptotic expansions of the displacement elastodynamic integral in terms of known functions
(Elsevier, 2025-05-01) 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
The integral [Formula presented] plays an essential role in the study of several phenomena in the theory of elastodynamics (Ceballos and Prato, 2014). But an exact evaluation of this integral in terms of known functions is not possible. In this paper, we derive an analytic representation of this integral in the form of convergent series of elementary functions and hypergeometric functions. This series have an asymptotic character for either, small values of the variable s, or for small values of the variables r and R. It is derived by using the asymptotic technique designed in Lopez (2008) for Mellin convolution integrals. Some numerical experiments show the accuracy of the approximation supplied by the first few terms of the expansion.
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
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.