The triangle-to-square transformation for finite-part integrals

1992 ◽  
pp. 72-78
Author(s):  
Ralf Kieser
Keyword(s):  
Finite Part

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

scholarly journals ZH production in gluon fusion: two-loop amplitudes with full top quark mass dependence

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Long Chen ◽  
Gudrun Heinrich ◽  
Stephen P. Jones ◽  
Matthias Kerner ◽  
Jonas Klappert ◽  
...  
Keyword(s):  
Top Quark ◽  
Quark Mass ◽  
Gluon Fusion ◽  
Top Quarks ◽  
Mass Dependence ◽  
Finite Part ◽  
Top Quark Mass ◽  
Helicity Amplitudes ◽  
Quark Mass Dependence ◽  
Loop Amplitudes

Abstract We present results for the two-loop helicity amplitudes entering the NLO QCD corrections to the production of a Higgs boson in association with a Z -boson in gluon fusion. The two-loop integrals, involving massive top quarks, are calculated numerically. Results for the interference of the finite part of the two-loop amplitudes with the Born amplitude are shown as a function of the two kinematic invariants on which the amplitudes depend.

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.

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