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In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ∫ 0 b x α f (x) Ai (− ω x) d x over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when ω ≫ 1 . The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [ 0 , 1 ] and [ 1 , b ] . For integrals over the interval [ 0 , 1 ] , we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [ 1 , b ] , we transform the Airy function into the first kind of Bessel function. By applying Cauchy's integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [ 0 , + ∞) , which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method. [ABSTRACT FROM AUTHOR]

This study focuses on the efficient and precise computation of Bessel transforms, defined as ∫abf(x)Jν(ωx)dx. Exploiting the integral representation of Jν(ωx), these Bessel transformations are reformulated into the oscillatory integrals of Fourier-type. When a>0, these Fourier-type integrals are transformed through distinct complex integration paths for cases with b

Based on Chebyshev spectral collocation and numerical techniques for handling highly oscillatory integrals, we propose a numerical method for a class of highly oscillatory Volterra integral equations frequently encountered in engineering applications. Specifically, we interpolate the unknown function at Chebyshev points, and substitute these points into the integral equation, resulting in a system of linear equations. The highly oscillatory integrals are treated using either the numerical steepest descent method or the Filon-Clenshaw-Curtis method. Additionally, we derive an error estimation formula for this method using error analysis techniques and validate the convergence and effectiveness of the proposed approach through numerical examples.

We study the efficient approximation of highly oscillatory integrals using Filon methods. A crucial step in the implementation of these methods is the accurate and fast computation of the Filon quadrature moments. In this work we demonstrate how recurrences can be constructed for a wide class of oscillatory kernel functions, based on the observation that many physically relevant kernel functions are in the null space of a linear differential operator whose action on the Filon interpolation basis is represented by a banded (infinite) matrix. We discuss in further detail the application to two classes of particular interest: integrals with algebraic singularities and stationary points and integrals involving a Hankel function. We provide rigorous stability results for the moment computation for the first of these classes and demonstrate how the corresponding Filon method results in an accurate approximation at truly frequency-independent cost. For the Hankel kernel, we derive error estimates that describe the convergence behaviour of the method in terms of frequency and number of Filon quadrature points. Finally, we show how Filon methods with recursive moment computation can be applied to efficiently compute integrals arising in hybrid numerical-asymptotic collocation methods for high-frequency wave scattering on a screen. [ABSTRACT FROM AUTHOR]

An antenna synthesis application is presented by solving a highly oscillatory Fourier integral using a stable and accurate Levin's algorithm. In antenna synthesis, the current distribution is obtained by the inverse Fourier integral of the antenna radiation pattern. Since this integral is highly oscillatory, the Levin method can be used for its solution. However, when the number of nodes or the frequency increases, the Levin method becomes unstable and ineffective due to the large condition number of the interpolation matrix. Thus, an improved scheme of the method is used in an antenna synthesis application in which reproducing kernel functions are used as the basis of the approximation function. The accuracy of the new method is verified by a log-periodic antenna example. The error and stability analysis results show that the new method is more stable and accurate than other well-known kernels, especially for a large number of nodes. [ABSTRACT FROM AUTHOR]

Using the Cauchy integral theorem, we develop the application of the steepest descent method to efficiently compute the three-dimensional acoustic single-layer integral operator for large wave numbers. Explicit formulas for the splitting points are derived in the one-dimensional case to build suitable complex integration paths. The construction of admissible steepest descent paths is next investigated and some of their properties are stated. Based on these theoretical results, we derive the quadrature scheme of the oscillatory integrals first in dimension one and then extend the methodology to three-dimensional planar triangles. Numerical simulations are finally reported to illustrate the accuracy and efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ∫0bxαf(x)Ai(−ωx)dx over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when ω≫1. The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [0,1] and [1,b]. For integrals over the interval [0,1], we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [1,b], we transform the Airy function into the first kind of Bessel function. By applying Cauchy’s integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [0,+∞), which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method.

We consider the highly oscillatory integral F(w):=∫−∞∞eiw(tK+2+eiθtp)g(t)dt$F(w):=\int _{-\infty }^\infty e^{iw(t^{K+2}+e^{i\theta }t^p)}g(t)dt$ for large positive values of w, −π<θ≤π$-\pi <\theta \le \pi$, K and p positive integers with 1≤p≤K$1\le p\le K$, and g(t)$g(t)$ an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when w→+∞$w\rightarrow +\infty$ for general values of K and p in terms of elementary functions, and determine the Stokes lines. For p≠1$p\ne 1$, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters K and p; the special case p=1$p=1$ requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals ΨK(x1,x2,...,xK)$\Psi _K(x_1,x_2,\ldots ,x_K)$ for large values of one of its variables, say xp$x_p$, and bounded values of the remaining ones. This family of integrals may be written in the form F(w)$F(w)$ for appropriate values of the parameters w, θ and the function g(t)$g(t)$. Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large |xp|$\vert x_p\vert$. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al. [ABSTRACT FROM AUTHOR]

This paper appertains the presentation of a Clenshaw–Curtis rule to evaluate highly oscillatory Fredholm integro-differential equations (FIDEs) with Cauchy and weak singularities. To calculate the singular integral, the unknown function approximated by an interpolation polynomial is rewritten as a Taylor series expansion. A system of linear equations of FIDEs obtained by using equally spaced points as collocation points is solved to obtain the unknown function. The proposed method attains higher accuracy rates, which are proven by error analysis and some numerical examples as well. [ABSTRACT FROM AUTHOR]

We present an efficient numerical algorithm for approximating integrals with highly oscillatory Hankel kernels. First, using the integral expression of the Hankel function, the integral is transformed into the integral of the trigonometric kernel, and the singularities are separated. According to whether the integration interval contains zeros or not, the integrals are divided into two cases: singular highly oscillatory integrals and non-singular highly oscillatory integrals. For the non-singular case, the Clenshaw–Curtis–Filon rules (CCF) method is used for fast calculation. For the second case, by using the Cauchy integral theorem, the integral is transformed into an infinite integral, which can be calculated by generalized Gaussian–Laguerre rules. The efficiency and accuracy of the new method are verified by numerical examples.

The information of earth structure is curial for designing the grounding system and evaluating the transmission line parameters. In order to obtain the earth structure using the measured data, an efficient algorithm for the parameter estimation of a horizontally multilayered soil by using a novel integration method is proposed. In the proposed algorithm, an adaptive iteration process will terminate till the predefined error reaches the requirement, where the earth parameters, such as the layer depth, layer resistivity, and layer number, are adjusted accordingly with the help of the Levenberg-Marquardt algorithm. In each iteration, the key is to evaluate the apparent resistivity and its partial derivatives with respect to soil parameters, which are expressed in the form of general Sommerfeld integral (GSI). In this paper, the GSI is divided into two parts. The first part with an infinite upper limit is integrated along a novel complex integration path instead of the conventional real axis path, which makes the evaluation of the GSI highly efficient due to the fast damping property of the transformed kernel. The second part with the finite integral interval is calculated using the cubic spline technique to have a semi-analytic expression, which can avoid evaluating the oscillatory kernel. In comparison with the results in the published literatures, numerical examples show that the proposed soil parameters estimation approach can reach good accuracy, and is robust and efficient. The reason is simple as the GSIs are calculated by using the well-designed and sophisticated approach. With extensive numerical experiments, the proposed algorithm is believed to be a capable and applicable candidate in engineering to invert the information of the earth structure.

This paper presents a method that is based on the sum of line integrals for fast computation of singular and highly oscillatory integrals ∫ c d G (x) e i μ (x − c) k d x , − ∞ > c > d > ∞ , and ∫ − 1 1 f (x) H l (x) e i μ x d x , l = 1 , 2 , 3. Where G and f are non-oscillatory sufficiently smooth functions on the interval of integration. H l is a product of singular factors and μ ≫ 1 is an oscillatory parameter. The computation of these integrals requires f and G to be analytic in a large complex region C accommodating the interval of integration. The integrals are changed into a problem of integrals on [ 0 , ∞) ; which are later computed using the generalized Gauss-Laguerre rule or by the construction of Gauss rules relative to a Freud weights function e − x k with k positive. MATHEMATICA programming code, algorithms and illustrative numerical examples are provided to test the efficiency of the presented experiments. [ABSTRACT FROM AUTHOR]

This paper presents an efficient method for the computation of two highly oscillatory integrals having logarithmic and Cauchy-logarithmic singularities. This approach first requires the transformation of the original oscillatory integrals into a sum of line integrals with semi-infinite intervals. Afterwards, the coefficients of the three-term recurrence relation that satisfy the orthog- onal polynomial are obtained by using the method based on moments, where classical Laguerre and Gautschi's logarithmic weight functions are employed. The algorithm reveals that with fixed n, the method is capable of achieving significant figures within a short time. Furthermore, the approach yields higher accuracy as the frequency increases. The results of numerical experiments are given to substantiate our theoretical analysis. [ABSTRACT FROM AUTHOR]

A method based on modification of numerical steepest descent method to efficiently compute highly oscillatory integrals having endpoint singularities of algebraic and logarithmic type is proposed in this paper. The three-term recursion coefficients for orthogonal polynomials with respect to Gautschi's weight function w G (t ; s) = t s (t − 1 − log t) e − t (s > − 1) on (0 , ∞) , as well as the corresponding quadrature formulas of Gaussian type, are used in this method. Finally, in order to illustrate the efficiency of the presented method a few numerical examples are included. The obtained results show that the proposed method is very efficient and economical in terms of computation time. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the numerical evaluation of hypersingular finite-part integrals with two kinds of highly oscillatory integrands. Suppose that f is analytic in the first quadrant of the complex plane, based on complex integration theory, both of them are transformed into the problem of integrating two integrals on [ 0 , + ∞) , such that the integrands do not oscillate and decay exponentially and thus can be computed efficiently by constructing the corresponding Gaussian quadrature rule for them. Moreover, error analyses are made for the proposed methods. Finally, several numerical examples are given to verify the theoretical results and illustrate the accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

A principal-value integral in wavenumber appears in the description of water waves generated by a wavemaker in wave tank. Its integrand is highly oscillatory and has a pole at the prescribed wavenumber associated with the oscillation frequency of wavemaker. Our analysis starts with transforming the wavenumber integral into two canonic integrals by changing the integral variable. The canonic integrals are then studied in three complex planes. Three contour integrals along appropriate steepest descent paths yield several components including residues' contribution and integral terms. By regrouping integral terms according to their integrand, a new formulation is derived and composed of an explicit steady-state component, an explicit initial component, and two integrals. The wavefront component in integral form is analysed and expressed in a close form involving the complementary error function. The local component is further expressed as the sum of an asymptotic term and remaining integral which is well suited for numerical computations. Numerical results confirm that the new formulation gives exactly the same as the original one but in a much more efficient and accurate way. Furthermore, the new formulation providing explicit expressions of different components and could be used to reveal insightful features of transient waves. [ABSTRACT FROM AUTHOR]

Abstract This paper explores the fast multipole methods (FMMs) to accelerate the approximation for weakly singular Volterra integral equations with highly oscillatory trigonometric kernels. By constructing the fast translation path, the FMM is utilized to speed up the iterative method, which reduces the complexity from O (N 2) to O (N). Especially, we use the collocation method to discretize the Volterra integral equation with constants and linear elements respectively, then apply the GMRES to solve the dense and non-symmetric linear system. In addition, the highly oscillatory integrals derived from the algorithm are calculated effectively by the steepest descent method. The proposed method shows that the numerical solutions become more accurate as the frequency increases. Both of the optimal convergence rates of truncation and the error bounds analysis are represented in the end. [ABSTRACT FROM AUTHOR]

The Filon-Clenshaw-Curtis method (FCC) for the computation of highly oscillatory integrals is known to attain surprisingly high precision. Yet, for large values of frequency $$\omega $$ it is not competitive with other versions of the Filon method, which use high derivatives at critical points and exhibit high asymptotic order. In this paper we propose to extend FCC to a new method, FCC $$+$$ , which can attain an arbitrarily high asymptotic order while preserving the advantages of FCC. Numerical experiments are provided to illustrate that FCC $$+$$ shares the advantages of both familiar Filon methods and FCC, while avoiding their disadvantages. [ABSTRACT FROM AUTHOR]

In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, ⨍ − 1 1 f ( x ) log ( x − α ) e i k x x − t d x , t ∉ ( − 1 , 1 ) , α ∈ [ − 1 , 1 ] for a smooth function f ( x ) . This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples.

In this paper, we present an exact integration scheme to compute highly oscillatory integrals that appear in the solution of the two-dimensional Helmholtz problem using the planewave-enriched partition of unity finite element method. In the proposed scheme, such oscillatory integrals are computed by a recursive application of the divergence theorem, eventually expressing the integrals in terms of evaluations of the corresponding integrands at the nodes of the finite element mesh. The number of such function evaluations is independent of the wave number k , which permits the scheme to be used for arbitrary high values of k . We consider finite element meshes with unstructured triangular and structured rectangular elements, and present numerical results for three canonical benchmark Helmholtz problems to demonstrate the accuracy and efficacy of the method. [ABSTRACT FROM AUTHOR]

Developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering. In this paper, we explore the regularization method for the coupled elliptic sine–Gordon equations along with Cauchy data. The system of equations originates from the static case of the coupled hyperbolic sine–Gordon equations modeling the coupled Josephson junctions in superconductivity, and so far it addresses the Josephson π -junctions. In general, the Cauchy problem is not well-posed, and herein the Hadamard-instability occurs drastically. Generalizing the kernel-based regularization method, we propose a stable approximate solution. Confirmed by the error estimate, this solution strongly converges to the exact solution in L 2 -norm. The main concern of this paper is also with the way to compute the regularized solution formed by an alike integral equation. We employ the proposed techniques that successfully approximated the highly oscillatory integral, and apply the Picard-like iteration to organize an efficient and reliable tool of computations. The results are viewed as the improvement as well as the generalization of many previous works. The paper is also accompanied by a numerical example that demonstrates the potential of this idea. [ABSTRACT FROM AUTHOR]

• Using integration by parts and the characteristics of Chebyshev polynomials, four useful recursive relationships of the required modified moments are deduced. • We perform the strict error analysis on the presented method and acquire asymptotic error estimations in inverse powers of frequency ω. • Our method has the following advantages: when the interpolation node is fixed, the accuracy improves considerably as either the frequency or the interpolated multiplicities at endpoints increase. • Our proposed method is simple to construct, the algorithm is easy to implement and has less running times. Moreover, only a small number of interpolation nodes is needed to achieve high accuracy. Our proposed method is applicable to many kinds of highly oscillatory Fourier-type integrals. This fully shows the wide application of this method. In this article, we propose and analyze the Clenshaw–Curtis–Filon-type method for computing many highly oscillatory Fourier-type integrals with algebraic or logarithmic singularities at the endpoints. First, by using integration by parts and the characteristics of Chebyshev polynomials, four useful recursive relationships of the required modified moments are deduced. We perform the strict error analysis on the presented method and acquire asymptotic error estimations in inverse powers of frequency ω. Our method has the following advantages: when the interpolation node is fixed, the accuracy improves considerably as either the frequency or the interpolated multiplicities at endpoints increase. Numerical experiments verify the efficacy and correctness of the presented method. [ABSTRACT FROM AUTHOR]

We propose a new method for the efficient approximation of a class of highly oscillatory weighted integrals where the oscillatory function depends on the frequency parameter , typically varying in a large interval. Our approach is based, for a fixed but arbitrary oscillator, on the pre-computation and low-parametric approximation of certain ω-dependent prototype functions whose evaluation leads in a straightforward way to recover the target integral. The difficulty that arises is that these prototype functions consist of oscillatory integrals which makes them difficult to evaluate. Furthermore, they have to be approximated typically in large intervals. Here we use the quantized-tensor train (QTT) approximation method for functional M-vectors of logarithmic complexity in M in combination with a cross-approximation scheme for TT tensors. This allows the accurate approximation and efficient storage of these functions in the wide range of grid and frequency parameters. Numerical examples illustrate the efficiency of the QTT-based numerical integration scheme on various examples in one and several spatial dimensions. [ABSTRACT FROM AUTHOR]

This paper discusses the computation of multi-dimensional highly oscillatory integrals. The approach here denoted by 'L-eRPIM' adopts the Levin collocation method and the enriched type radial point interpolation method (eRPIM) appropriately, showing improvement in accuracy as the frequency increases. In this approach a multi-dimensional highly oscillatory integral is first converted into a partial differential equation (PDE) by Levin collocation method. Then by eRPIM the solution of the resulting PDE is obtained to compute the value of integral. As the advantages of this approach we can mention that the well known results concerning the order of error in terms of frequency ω and dimension d are still valid and sensitivity to the shape parameter of the RBFs can also be decreased by adding more monomial and trigonometric basis functions. Numerical and practical examples are presented to demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: In this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities. For these quadrature rules we derive error estimates in terms of , the number of nodes, the rate of oscillations and a Sobolev-like regularity of the function. We prove that the method is not only robust but the error even decreases, for fixed , as increases. Practical issues about the implementation of the rule are also covered in this paper by: (a) writing down ready-to-implement algorithms; (b) analysing the numerical stability of the computations and (c) estimating the overall computational cost. We finish by showing some numerical experiments which illustrate the theoretical results presented in this paper. [Copyright &y& Elsevier]

An adaptive integration algorithm is presented for the computation of the Physical Optics (PO) electric and magnetic field scattered by electrically large objects modeled by Non-Uniform Rational B-Splines (NURBS). The algorithm is the customization of a more general-purpose result that has been recently published. By using a unique formulation both impenetrable (e.g., impedance surfaces, coated conductors) as well as transparent thin sheet materials (e.g., thin dielectric panels, or frequency selective surfaces) are treated, via their Fresnel reflection and transmission coefficients. The PO radiation integral is evaluated over the NURBS parametric domain. Since most of the computer-aided geometric design (CAGD) tools are based on NURBS, the proposed algorithm allows a straightforward electromagnetic analysis of the structures by exploiting the standard available geometrical description, with no need of generating new geometrical models. Furthermore, the proposed adaptive sampling requires a number of integration points that is found to be drastically smaller than that resulting from standard Nyquist-based sampling integration algorithms. Such reduction of the sampling points is achieved by resorting to high-frequency technique concepts and allows a significant reduction of the CPU computational burden. Therefore the algorithm is efficient and particulary suitable for the electromagnetic characterization of real-life electrically large objects. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we present a new quadrature method for semi-infinite range highly oscillatory integrals with integrands of the form , where the phase function g and its derivative are positive, unboundedly increasing on a subinterval of the integration interval. The method is based on approximating 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 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. [Copyright &y& Elsevier]

In this paper we obtain new results on Filon-type methods for computing oscillatory integrals of the form ∫−11f(s)exp(iks)ds. We use a Filon approach based on interpolating f at the classical Clenshaw–Curtis points cos(jπ/N),j=0,…,N. The rule may be implemented in O(NlogN) operations. We prove error estimates that show explicitly how the error depends both on the parameters k and N and on the Sobolev regularityof f. In particular we identify the regularity of f required to ensure the maximum rate of decay of the error as k → ∞. We also describe a method for implementing the method and prove its stability both when N ≤ k and N > k. Numerical experiments illustrate both the stability of the algorithm and the sharpness of the error estimates. [ABSTRACT FROM PUBLISHER]

Abstract: Filon-type methods for computing highly oscillatory integrals with an algebraic singularity of the form , where β >0, α + β +1>0 and f is a sufficiently smooth function on [0,1] and ω ≫1, has been proposed by Hascelik [A.I. Hascelik, Suitable Gauss and Filon-type methods for oscillatory integrals with an algebraic singularity, Appl. Numer. Math 59 (2009) 101–118]. In this paper, we first expand such integrals into asymptotic series in inverse powers of ωβ and then give the asymptotic order of the Filon-type methods. Numerical examples are provided to confirm our analysis. [Copyright &y& Elsevier]

Abstract: We analyze the convergence behavior of Filon-type quadrature rules by making explicit the dependence on both , the parameter that controls the oscillatory behavior of the integrand, and , the number of function evaluations. We provide explicit conditions on the domain of analyticity of the integrand to ensure convergence for . [Copyright &y& Elsevier]

A new technique to compute the physical optics (PO) integral is presented. The technique consists of a blind code that computes the different contributions (stationary phase points, end points, etc.) numerically. This technique is based on a decomposition of the surface into small triangles and a fast evaluation of each triangle by means of a deformation of the integration path in the complex plane. This algorithm permits a fast and accurate evaluation of the PO integral for smooth large surfaces. The CPU time is almost independent of frequency. [ABSTRACT FROM AUTHOR]

We investigate using the GMRES method with the differentiation operator. This operator is unbounded and thus does not fall into the framework of existing Krylov subspace theory. We establish conditions under which a function can be approximated by its own derivatives in a domain of the complex plane. These conditions are used to determine when GMRES converges. This algorithm outperforms traditional quadrature schemes for a large class of highly oscillatory integrals, even when the kernel of oscillations is unknown. [ABSTRACT FROM AUTHOR]

Wave-based numerical methods often require to integrate products of polynomials and exponentials. With quadrature methods, this task can be particularly expensive at high frequencies as large numbers of integration points are required. This paper presents a set of closed-form solutions for the integrals of polynomial–exponential products in two and three dimensions. These results apply to arbitrary polygons in two dimensions, and for arbitrary polygonal surfaces or polyhedral volumes in three dimensions. Quadrature methods are therefore not required for this class of integrals that can be evaluated quickly and exactly. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: The standard classic integration rules give inaccurate results for where , are real numbers and f is any sufficiently smooth function on . These integrals have been investigated for the special case in Hascelik [A.I. Hascelik, On numerical computation of integrals with integrands of the form on (2007), in press] and for the case (, ) in Gautschi [W. Gautschi, Computing polynomials orthogonal with respect to densely oscillating and exponentially decaying weight functions and related integrals, J. Comput. Appl. Math. 184 (2005) 493–504]. In this work we construct suitable Gauss quadrature rules for approximating these integrals in high accuracy. The required three-term recurrence coefficients are computed by the Chebyshev algorithm using arbitrary precision arithmetic. We also give appropriate Filon-type methods for these integrals, with related error bounds. Some numerical examples are given to test the new methods. [Copyright &y& Elsevier]

Abstract: With existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the form where is a sufficiently smooth function on [0, 1]. Gautschi has developed software (as scripts in Matlab) for computing these integrals for the special case . In this paper, an algorithm (as a Mathematica program) is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range. Numerical examples are given of testing the performance of the algorithm/program. [Copyright &y& Elsevier]