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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 : ...
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 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 ...
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 ...
Convergent expansions of the Bessel functions in terms of elementary functions
(Springer US, 2018)
info:eu-repo/semantics/article,
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
(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 ...
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. ...