The double exponential formula for oscillatory functions over the half infinite interval
ISCM '90 Proceedings of the International Symposium on Computation mathematics
Fast integration of rapidly oscillatory functions
Journal of Computational and Applied Mathematics
The Mathematica book (4th edition)
The Mathematica book (4th edition)
A Modification of Filon's Method of Numerical Integration
Journal of the ACM (JACM)
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation
SIAM Journal on Numerical Analysis
Evaluating infinite range oscillatory integrals using generalised quadrature methods
Applied Numerical Mathematics
On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]
Journal of Computational and Applied Mathematics
Suitable Gauss and Filon-type methods for oscillatory integrals with an algebraic singularity
Applied Numerical Mathematics
Shifted GMRES for oscillatory integrals
Numerische Mathematik
Journal of Computational and Applied Mathematics
GMRES for the Differentiation Operator
SIAM Journal on Numerical Analysis
An improved Levin quadrature method for highly oscillatory integrals
Applied Numerical Mathematics
Asymptotic Analysis of Numerical Steepest Descent with Path Approximations
Foundations of Computational Mathematics
Exponentially-fitted Gauss-Laguerre quadrature rule for integrals over an unbounded interval
Journal of Computational and Applied Mathematics
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In this paper, we present a new quadrature method for semi-infinite range highly oscillatory integrals with integrands of the form f(x)exp[i@wg(x)], where the phase function g and its derivative are positive, unboundedly increasing on a subinterval [c,~] of the integration interval. The method is based on approximating f/g^' by a linear combination of negative rational powers of the phase function so that the moments can be expressed by the extended exponential integral function. If the magnitude of @wg^'(c) is sufficiently large, our method is very efficient in obtaining very high precision approximations to the integral, without computation of derivatives or the inverse of the phase function. The effectiveness of the method is discussed in the light of a set of test examples including the first problem of the SIAM 100-Digit Challenge, the Bessoid integral, and two finite range integrals. We also present a Mathematica program to be used for the implementation of the method.