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Convergent expansions of the incomplete gamma functions in terms of elementary functions
(World Scientific Publishing, 2017)
info:eu-repo/semantics/article,
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 ...
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 ...
Zernike-like systems in polygons and polygonal facets
(Optical Society of America, 2015)
info:eu-repo/semantics/article,
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 ...
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 ...
Convergent and asymptotic expansions of the Pearcey integral
(Elsevier, 2015)
info:eu-repo/semantics/article,
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 ...
The Pearcey integral in the highly oscillatory region
(Elsevier, 2016)
info:eu-repo/semantics/article,
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. ...
Analytic formulas for the evaluation of the Pearcey integral
(American Mathematical Society, 2017)
info:eu-repo/semantics/article,
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 ...
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 ...
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 ...