Now showing items 1-20 of 20

    • Analytic formulas for the evaluation of the Pearcey integral 

      López García, José Luis; Pagola Martínez, Pedro Jesús (American Mathematical Society, 2017)   Artículo / Artikulua  OpenAccess
      We can find in the literature several convergent and/or asymptotic expansions of the Pearcey integral P(x, y) in different regions of the complex variables x and y, but they do not cover the whole complex x and y planes. The ...
    • Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter 

      Ferreira, Chelo; López García, José Luis; Pérez Sinusía, Ester (Shanghai Normal UniversityWilmington Scientific Publisher, 2018)   Artículo / Artikulua  OpenAccess
      In previous papers [6–8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not ...
    • The asymptotic expansion of the swallowtail integral in the highly oscillatory region 

      Ferreira, Chelo; López García, José Luis; Pérez Sinusía, Ester (Elsevier, 2018)   Artículo / Artikulua
      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 ...
    • Convergent and asymptotic expansions of the Pearcey integral 

      López García, José Luis; Pagola Martínez, Pedro Jesús (Elsevier, 2015)   Artículo / Artikulua  OpenAccess
      We consider the Pearcey integral P(x; y) for large values of |x|, x, y ∈ C. We can find in the literature several convergent or asymptotic expansions in terms of elementary and special functions, with different levels of ...
    • Convergent and asymptotic methods for second-order difference equations with a large parameter 

      Ferreira, Chelo; López García, José Luis; Pérez Sinusía, Ester (Springer, 2018)   Artículo / Artikulua
      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 ...
    • Convergent asymptotic expansions of Charlier, Laguerre and Jacobi polynomials 

      López García, José Luis; Temme, Nico M. (Cambridge University Press, 2004)   Artículo / Artikulua  OpenAccess
      Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of ...
    • Convergent expansions of the Bessel functions in terms of elementary functions 

      López García, José Luis; Pagola Martínez, Pedro Jesús (Springer US, 2018)   Artículo / Artikulua  OpenAccess
      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 ...
    • Convergent expansions of the incomplete gamma functions in terms of elementary functions 

      Bujanda Cirauqui, Blanca Upna; López García, José Luis; Pagola Martínez, Pedro Jesús (World Scientific Publishing, 2017)   Artículo / Artikulua  OpenAccess
      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 ...
    • An extension of the multiple Erdélyi-Kober operator and representations of the generalized hypergeometric functions 

      Karp, B.D.; López García, José Luis (De Gruyter, 2018)   Artículo / Artikulua
      In this paper we investigate the extension of the multiple Erd elyi-Kober fractional integral operator of Kiryakova to arbitrary complex values of parameters by the way of regularization. The regularization involves ...
    • Formulas for the amplitude of the van der Pol limit cycle through the homotopy analysis method 

      López García, José Luis; Abbasbandy, S.; López Ruiz, R. (Hindawi / Wiley, 2009)   Artículo / Artikulua  OpenAccess
      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 ...
    • Generalization of Zernike polynomials for regular portions of circles and ellipses 

      Navarro, Rafael; López García, José Luis; Díaz, José A.; Pérez Sinusía, Ester (Optical Society of America, 2014)   Artículo / Artikulua  OpenAccess
      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 ...
    • A note on the asymptotic expansion of the Lerch’s transcendent 

      Cai, Xing Shi; López García, José Luis (Taylor & Francis, 2018)   Artículo / Artikulua  OpenAccess
      In Ferreira and López [Asymptotic expansions of the Hurwitz–Lerch zeta function. J Math Anal Appl. 2004;298(1):210–224], the authors derived an asymptotic expansion of the Lerch's transcendent Φ(z,s,a) for large |a|, valid ...
    • On a modifcation of Olver's method: a special case 

      Ferreira, Chelo; López García, José Luis; Pérez Sinusía, Ester (Springer US, 2016)   Artículo / Artikulua  OpenAccess
      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 ...
    • Orthogonal basis with a conicoid first mode for shape specification of optical surfaces 

      Ferreira, Chelo; López García, José Luis; Navarro, Rafael; Pérez Sinusía, Ester (Optical Society of America, 2016)   Artículo / Artikulua  OpenAccess
      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 ...
    • The Pearcey integral in the highly oscillatory region 

      López García, José Luis; Pagola Martínez, Pedro Jesús (Elsevier, 2016)   Artículo / Artikulua  OpenAccess
      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. ...
    • A simplification of the stationary phase method: application to the Anger and Weber functions 

      López García, José Luis (Kent State UniversityJohann Radon Institute (RICAM), 2017)   Artículo / Artikulua  OpenAccess
      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 ...
    • Uniform convergent expansions of the Gauss hypergeometric function in terms of elementary functions 

      Ferreira, Chelo; López García, José Luis; Pérez Sinusía, Ester (Taylor & Francis, 2018)   Artículo / Artikulua  OpenAccess
      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 ...
    • Uniform representation of the incomplete beta function in terms of elementary functions 

      Ferreira, Chelo; López García, José Luis; Pérez Sinusía, Ester (Kent State UniversityJohann Radon Institute (RICAM), 2018)   Artículo / Artikulua  OpenAccess
      We consider the incomplete beta function Bz(a, b) in the maximum domain of analyticity of its three variables: a, b, z ∈ C, −a /∈ N, z /∈ [1, ∞). For <b ≤ 1 we derive a convergent expansion of z−aBz(a, b) in terms of the ...
    • The use of two-point Taylor expansions in singular one-dimensional boundary value problems I 

      Ferreira, Chelo; López García, José Luis; Pérez Sinusía, Ester (Elsevier, 2018)   Artículo / Artikulua
      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 ...
    • Zernike-like systems in polygons and polygonal facets 

      Ferreira, Chelo; López García, José Luis; Navarro, Rafael; Pérez Sinusía, Ester (Optical Society of America, 2015)   Artículo / Artikulua  OpenAccess
      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 disk. In [Diaz et all, 2014] we introduced a new Zernike ...