MOVABLE SINGULARITIES AND QUADRATURE 285 Proof
نویسندگان
چکیده
A general procedure is described for treating a movable singularity in an integral. This enables us to change the original integral I0 into GI¡, where G depends only on the parameters of the singularity and /, is a new integral which exists for all values of the parameters. The results are then applied to the particular problem of evaluating f1 f(x)dx .,{(!x2Xl fc2x2)}"2' where / is entire and k varies between 0 and 1. Some new quadrature schemes and new effective methods of evaluating incomplete elliptic integrals are derived.
منابع مشابه
Painlevé analysis and normal forms theory
Nonlinear vector fields have two important types of singularities: the fixed points in phase space and the time singularities in the complex plane. The first singularities are locally analyzed via normal form theory whereas the second ones are studied by the Painlevé analysis. In this paper, normal form theory is used to describe the solutions around their complex-time singularities. To do so, ...
متن کاملOptimal Adaptive Ridgelet Schemes for Linear Transport Equations
In this paper we present a novel method for the numerical solution of linear transport equations, which is based on ridgelets. Such equations arise for instance in radiative transfer or in phase contrast imaging. Due to the fact that ridgelet systems are well adapted to the structure of linear transport operators, it can be shown that our scheme operates in optimal complexity, even if line sing...
متن کاملOn Generalized Gaussian Quadrature Rules for Singular and Nearly Singular Integrals
We construct and analyze generalized Gaussian quadrature rules for integrands with endpoint singularities or near endpoint singularities. The rules have quadrature points inside the interval of integration and the weights are all strictly positive. Such rules date back to the study of Chebyshev sets, but their use in applications has only recently been appreciated. We provide error estimates an...
متن کاملExtended Gaussian quadratures for functions with an end-point singularity of logarithmic type
The extended Gaussian quadrature rules are shown to be an efficient tool for numerical integration of wide class of functions with singularities of logarithmic type. The quadratures are exact for the functions pol1n−1(x) + lnx pol2n−1(x) , where pol1n−1(x) and pol2n−1(x) are two arbitrary polynomials of degree n−1 and n is the order of the quadrature formula. We present an implementation of num...
متن کاملVariable transformations and Gauss-Legendre quadrature for integrals with endpoint singularities
Gauss–Legendre quadrature formulas have excellent convergence properties when applied to integrals ∫ 1 0 f(x) dx with f ∈ C∞[0, 1]. However, their performance deteriorates when the integrands f(x) are in C∞(0, 1) but are singular at x = 0 and/or x = 1. One way of improving the performance of Gauss–Legendre quadrature in such cases is by combining it with a suitable variable transformation such ...
متن کامل