On ignoring the singularity in numerical quadrature

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Numerical Quadrature of Fourier Transform

In some cases the function <£(£) or \pik) is given by a closed expression which is too complicated to permit a sufficiently accurate analytic evaluation of the integral for the entire range of the parameter x. In other cases 4>ik) or ^(&) may be available only in numerical form. The conventional methods of numerical quadrature (e.g., Simpson's rule) are not suitable for evaluation of the above ...

Numerical Quadrature: Theory and Computation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Chapter

Numerical Quadrature and Asymptotic Expansions

On the computation of Macdonald functions by numerical quadrature

The use of Gaussian quadrature formulae is explored for the computation of the Macdonald function Kν(x) = ∫ ∞ 0 e cosh t cosh νt dt when x > 0 and ν is complex, ν = α+ iβ. It is shown that Gaussian quadrature with weight function w(t) = exp(−et) on [0,∞] is a viable approach, unless x is small and/or β large, but in combination with Gauss–Legendre quadrature, even in these latter cases.