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The use of two-point Taylor expansions in singular one-dimensional boundary value problems I
(Elsevier, 2018)
info:eu-repo/semantics/article,
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 ...
Asymptotic and convergent expansions for solutions of third-order linear differential equations with a large parameter
(Shanghai Normal UniversityWilmington Scientific Publisher, 2018)
info:eu-repo/semantics/article,
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 ...
Convergent and asymptotic methods for second-order difference equations with a large parameter
(Springer, 2018)
info:eu-repo/semantics/article,
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 expansion of the swallowtail integral in the highly oscillatory region
(Elsevier, 2018)
info:eu-repo/semantics/article,
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 ...
Uniform convergent expansions of the Gauss hypergeometric function in terms of elementary functions
(Taylor & Francis, 2018)
info:eu-repo/semantics/article,
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
(Kent State UniversityJohann Radon Institute (RICAM), 2018)
info:eu-repo/semantics/article,
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 ...
Convergent asymptotic expansions of Charlier, Laguerre and Jacobi polynomials
(Cambridge University Press, 2004)
info:eu-repo/semantics/article,
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 ...
On a modifcation of Olver's method: a special case
(Springer US, 2016)
info:eu-repo/semantics/article,
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 : ...
Orthogonal basis with a conicoid first mode for shape specification of optical surfaces
(Optical Society of America, 2016)
info:eu-repo/semantics/article,
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 ...
A simplification of the stationary phase method: application to the Anger and Weber functions
(Kent State UniversityJohann Radon Institute (RICAM), 2017)
info:eu-repo/semantics/article,
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 ...