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López García, José Luis

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López García

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José Luis

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Estadística, Informática y Matemáticas

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InaMat2. Instituto de Investigación en Materiales Avanzados y Matemáticas

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0000-0002-6050-9015

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2369

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Now showing 1 - 10 of 16
  • PublicationOpen 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.
  • PublicationOpen 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
  • PublicationOpen Access
    The uniform asymptotic method "saddle point near an end point" revisited
    (Elsevier, 2024) López García, José Luis; Pagola Martínez, Pedro Jesús; Palacios Herrero, Pablo; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertistate Publikoa
    We continue the program initiated in [López & all, 2009–2011] to simplify asymptotic methods for integrals: in this paper we revise the uniform method ‘‘saddle point near an end point’’. We obtain a more systematic version of this uniform asymptotic method where the computation of the coefficients of the asymptotic expansion is remarkably simpler than in the classical method. On the other hand, as in the standard method, the asymptotic sequence is given in terms of parabolic cylinder functions. New asymptotic expansions of the confluent hypergeometric functions 𝑀(𝑐, 𝑥∕𝛼 + 𝑐 + 1, 𝑥) and 𝑈(𝑐, 𝛼𝑥 + 𝑐 + 1, 𝑥) for large 𝑥, 𝑐 fixed, uniformly valid for 𝛼 ∈ (0, ∞), are given as an illustration.
  • PublicationOpen 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].
  • PublicationOpen Access
    Convergent expansions of the confluent hypergeometric functions in terms of elementary functions
    (American Mathematical Society, 2018) Bujanda Cirauqui, Blanca; López García, José Luis; Pagola Martínez, Pedro Jesús; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
    We consider the confluent hypergeometric function M(a, b; z) for z ∈ C and Rb >Ra > 0, and the confluent hypergeometric function U(a, b; z) for b ∈ C, Ra > 0, and Rz > 0. We derive two convergent expansions of M(a, b; z); one of them in terms of incomplete gamma functions γ(a, z) and another one in terms of rational functions of ez and z. We also derive a convergent expansion of U(a, b; z) in terms of incomplete gamma functions γ(a, z) and Γ(a, z). The expansions of M(a, b; z) hold uniformly in either Rz ≥ 0 or Rz ≤ 0; the expansion of U(a, b; z) holds uniformly in Rz > 0. The accuracy of the approximations is illustrated by means of some numerical experiments.
  • PublicationOpen 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.
  • PublicationOpen 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.
  • PublicationOpen 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.
  • PublicationOpen Access
    Convergent expansions of the Bessel functions in terms of elementary functions
    (Springer US, 2018) 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 Bessel functions Jν (z) and Yν (z) for ν > −1/2 and z ≥ 0. We derive a convergent expansion of Jν (z) in terms of the derivatives of (sin z)/z, and a convergent expansion of Yν (z) in terms of derivatives of (1−cos z)/z, derivatives of (1 − e−z)/z and (2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments.
  • PublicationOpen 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 Matemáticas
    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.