On the numerical evaluation of some definite integrals

1964 ◽  
Vol 60 (1) ◽  
pp. 75-82
Author(s):  
W. Barrett
Keyword(s):  
Numerical Evaluation ◽  
Substantial Reduction ◽  
Gaussian Quadrature ◽  
Contour Integral ◽  
Quadrature Formulae ◽  
Definite Integrals ◽  
Asymptotic Formulae

AbstractAn account is given of the application of Gaussian quadrature formulae to the numerical evaluation of certain definite integrals appearing in a recent paper by Craggs (2). Suitable changes of variable are used to effect a substantial reduction in the order of formula required. In comparing methods, use is made of an expression for the remainder as a contour integral, whose value is estimated with the aid of certain asymptotic formulae.

scholarly journals An Algorithm for the Numerical Evaluation of Certain Finite Part Integrals

2011 ◽  
Vol 2011 ◽  
pp. 1-21
Author(s):  
Samir A. Ashour ◽  
Hany M. Ahmed
Keyword(s):  
Numerical Evaluation ◽  
Gaussian Quadrature ◽  
Quadrature Rules ◽  
Finite Part ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Cauchy Principal Value Integrals ◽  
Principal Value

Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. In this work we present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals , and extend Clenshow's algorithm to evaluate these integrals in a stable way.

Nonstandard Gaussian quadrature formulae based on operator values

2009 ◽  
Vol 32 (4) ◽  
pp. 431-486 ◽  
Author(s):  
Gradimir V. Milovanović ◽  
Aleksandar S. Cvetković
Keyword(s):  
Gaussian Quadrature ◽  
Quadrature Formulae

Maximum of the modulus of kernels of Gaussian quadrature formulae for one class of Bernstein–Szegő weight functions

2012 ◽  
Vol 218 (9) ◽  
pp. 5746-5756 ◽  
Author(s):  
Miodrag M. Spalević ◽  
Miroslav S. Pranić ◽  
Aleksandar V. Pejčev
Keyword(s):  
Gaussian Quadrature ◽  
Weight Functions ◽  
Quadrature Formulae

On the variance of gaussian quadrature formulae in the ultraspherical case

CALCOLO ◽  
1994 ◽  
Vol 31 (1-2) ◽  
pp. 1-33
Author(s):  
K. J. Förster ◽  
K. Petras
Keyword(s):  
Gaussian Quadrature ◽  
Quadrature Formulae

scholarly journals A Note on a Fourier Sine Transform

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1828
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Integral Method ◽  
Contour Integral ◽  
Summary Table ◽  
Zero Distribution ◽  
Definite Integrals ◽  
Sine Transform ◽  
Fourier Sine ◽  
Fourier Sine Transform ◽  
General Power ◽  
Almost All

This is a compilation of definite integrals of the product of the hyperbolic cosecant function and polynomial raised to a general power. In this work, we used our contour integral method to derive a Fourier sine transform in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. A summary table of the results are produced for easy reading. A vast majority of the results are new.

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