Chelo Ferreira

An Asymptotic Expansion of the Double Gamma Function

The Barnes double gamma function G(z) is considered for large argument z. A new integral representation is obtained for logG(z). An asymptotic expansion in decreasing powers of z and uniformly valid for |Argz|<@p is derived from this integral. The expansion is accompanied by an error bound at any order of the approximation. Numerical experiments show that this bound is very accurate for real z. The accuracy of the error bound decreases for increasing Argz.

Asymptotic Expansions of Generalized Stieltjes Transforms of Algebraically Decaying Functions

Asymptotic expansions of Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative integer powers of their variable have been exhaustively investigated by R. Wong. In this article, we extend this analysis to Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative rational powers of their variable. Distributional approach is used to derive asymptotic expansions of the Stieltjes and generalized Stieltjes transforms of this kind of functions for large values of the parameter(s) of the transformation. Error bounds are obtained at any order of the approximation for a large family of integrands. The asymptotic approximation of an integral involved in the calculation of the mass renormalization of the quantum scalar field and of the third symmetric elliptic integral are given as illustrations.

Asymptotic relations in the Askey scheme for hypergeometric orthogonal polynomials

It has been recently pointed out that several orthogonal polynomials of the Askey table admit asymptotic expansions in terms of Hermite and Laguerre polynomials [Lopez and Temme, Meth. Appl. Anal. 6 (1999) 131-146; J. Comp. Appl. Math. 133 (2001) 623-633]. From those expansions, several known and new limits between polynomials of the Askey table were obtained in [Lopez and Temme, Meth. Appl. Anal. 6 (1999) 131-146; J. Comp. Appl. Math. 133 (2001) 623-633]. In this paper, we make an exhaustive analysis of the three lower levels of the Askey scheme which completes the asymptotic analysis performed in [Lopez and Temme, Meth. Appl. Anal. 6 (1999) 131-146; J. Comp. Appl. Math. 133 (2001) 623-633]: (i) We obtain asymptotic expansions of Charlier, Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Hermite polynomials. (ii) We obtain asymptotic expansions of Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Charlier polynomials. (iii) We give new proofs for the known limits between polynomials of these three levels and derive new limits.

Analytic expansions of thermonuclear reaction rates

The evaluation of thermonuclear reaction rates requires the calculation of several thermonuclear functions. These functions can be written as the Laplace transform of locally integrable functions which have an asymptotic expansion in negative rational powers of their variable. In this paper we obtain asymptotic expansions of the Laplace transform of these kinds of functions for small values of the parameter of the transformation. Error bounds are obtained at any order of the approximation for a large family of Laplace transforms which include thermonuclear functions. Then we apply this asymptotic theory to the calculation of convergent expansions of four thermonuclear functions in powers of the dimensionless Sommerfeld parameter. Some of these expansions also involve logarithmic terms in the dimensionless Sommerfeld parameter. Accurate error bounds are given at any order of the approximation.

Orthogonal basis with a conicoid first mode for shape specification of optical surfaces

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 different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed.

Asymptotic expansions of the Lauricella hypergeometric function F D

The Lauricella hypergeometric function FDr(a, b1,...,br;c;x1,...,xr) with r ∈ N, is considered for large values of one variable: x1, or two variables: x1 and x2. An integral representation of this function is obtained in the form of a generalized Stieltjes transform. Distributional approach is applied to this integral to derive four asymptotic expansions of this function in increasing powers of the asymptotic variable(s) 1 - x1 or 1 - x1 and 1 - x2. For certain values of the parameters a, bi and c, two of these expansions also involve logarithmic terms in the asymptotic variable(s). For large x1, coefficients of these expansions are given in terms of the Lauricella hypergeometric function FDr-1(a, b2,...,br;c;x2,...,xr) and its derivative with respect to the parameter a, whereas for large x1 and x2 those coefficients are given in terms of FDr-2(a, b3,...,br;c;x3,...,xr) and its derivative. All the expansions are accompanied by error bounds for the remainder at any order of the approximation. Numerical experiments show that these bounds are considerably accurate.

Two Algorithms for Computing the Randles–Sevcik Function from Electrochemistry

AbstractWe derive two expansions of the Randles–Sevcik function

Estimation of the relative sensitivity of qPCR analysis using pooled samples.

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