High-speed high-accuracy computation of an infinite integral with unbounded and oscillated integrand
نویسنده
چکیده
We propose an efficient computation method for an infinite integral ∞ 0 xdx/(1 + x 6 sin 2 x), which has an unbounded integrand with highly oscillated singularity. Computing the value of this integral has been a problem since 1984. We herein demonstrate that the method using the Hilbert transform to change this type of singular function into a smooth function and compute the value of the integral of one million or more significant digits using a superconvergent double exponential quadrature.
منابع مشابه
Numerical Evaluation of Goursat’s Infinite Integral with an Unbounded Function
The infinite integral ∫ ∞ 0 xdx/(1 + x 6 sin2 x) converges but is hard to evaluate because the integrand f (x) = x/(1 + x6 sin2 x) is a non-convergent and unbounded function, indeed f (kπ) = kπ → ∞ (k → ∞). We show an efficient method to evaluate the above integral in high accuracy and actually obtain an approximate value in up to 73 significant digits on an octuple precision system in C++.
متن کاملFast computation of Goursat's infinite integral with very high accuracy
We propose an efficient computation method for the infinite integral ∫∞ 0 xdx/(1+ x 6 sin x), whose integrand contains a series of spikes, approximately π apart, growing taller and narrower as x increases. Computing the value of this integral has been a problem since 1984. We herein demonstrate a method using the Hilbert transform for changing this type of singular function into a smooth functi...
متن کاملSolvability of infinite system of nonlinear singular integral equations in the C(Itimes I, c) space and modified semi-analytic method to find a closed-form of solution
In this article, we discuss about solvability of infinite systems of singular integral equations with two variables in the Banach sequence space $C(I times I, c)$ by applying measure of noncompactness and Meir-Keeler condensing operators. By presenting an example, we have illustrated our results. For validity of the results we introduce a modified semi-analytic method in the case of tw...
متن کاملAxisymmetric Scaled Boundary Finite Element Formulation for Wave Propagation in Unbounded Layered Media
Wave propagation in unbounded layered media with a new formulation of Axisymmetric Scaled Boundary Finite Element Method (AXI-SBFEM) is derived. Dividing the general three-dimensional unbounded domain into a number of independent two-dimensional ones, the problem could be solved by a significant reduction in required storage and computational time. The equations of the corresponding Axisymmetri...
متن کاملComputing diffraction integrals with the numerical method of steepest descent
A common type of integral to solve numerically in computational room acoustics and other applications is the diffraction integral. Various formulations are encountered but they are usually of the Fourier-type, which means an oscillating integrand which becomes increasingly expensive to compute for increasing frequencies. Classical asympotic solution methods, such as the stationary-phase method,...
متن کامل