Improvement of the asymptotic behaviour of the Euler–Maclaurin formula for Cauchy principal value and Hadamard finite-part integrals

2004 ◽  
Vol 61 (4) ◽  
pp. 496-513 ◽  
Author(s):  
U. Jin Choi ◽  
Shin Wook Kim ◽  
Beong In Yun
Keyword(s):  
Asymptotic Behaviour ◽  
Finite Part ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Hadamard Finite Part ◽  
Principal Value ◽  
Maclaurin Formula

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.

A Generalization of the Cauchy Principal Value

1957 ◽  
Vol 9 ◽  
pp. 110-117 ◽  
Author(s):  
Charles Fox
Keyword(s):  
Finite Part ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Improper Integral ◽  
Principal Value ◽  
Image Position

If a < u < b and n > 0 then(1)is a so-called improper integral owing to the infinity in the integrand at x = u. When n = 0 we have associated with (1) the well-known Cauchy principal value, namely(2).Hadamard (1, p. 117 et seq.) derives from an improper integral an expression which he calls its finite part and which, as he shows, possesses many important properties.

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