Algorithm 858

Algorithm 858: Computing infinite range integrals of an arbitrary product of Bessel functions. We present an algorithm to compute integrals of the form ∫∞0 xm ∏ki = 1Jνi(aix)dx with Jνi(x) the Bessel function of the first kind and (real) order νi. The parameter m is a real number such that ∑i νi + m > −1 and the coefficients ai are strictly positive real numbers. The main ingredients in this algorithm are the well-known asymptotic expansion for Jνi(x) and the observation that the infinite part of the integral can be approximated using the incomplete Gamma function Γ(a,z). Accurate error estimates are included in the algorithm, which is implemented as a MATLAB program.

This software is also peer reviewed by journal TOMS.

References in zbMATH (referenced in 13 articles )

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  1. Oliveira de Silva, Diogo; Quilodrán, René: Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres (2021)
  2. Sugiura, Hiroshi; Hasegawa, Takemitsu: A truncated Clenshaw-Curtis formula approximates integrals over a semi-infinite interval (2021)
  3. Bornemann, Folkmar: The SIAM 100-Digit Challenge: a decade later. Inspirations, ramifications, and other eddies left in its wake (2016)
  4. Ratnanather, J. Tilak; Kim, Jung H.; Zhang, Sirong; Davis, Anthony M. J.; Lucas, Stephen K.: Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions (2014)
  5. Ibe, Oliver C.: Elements of random walk and diffusion processes (2013)
  6. Mehrem, Rami: Analytical evaluation of an infinite integral over four spherical Bessel functions (2013)
  7. Sidi, Avram: A user-friendly extrapolation method for computing infinite range integrals of products of oscillatory functions (2012)
  8. Gil, Amparo; Segura, Javier; Temme, Nico M.: Basic methods for computing special functions (2011)
  9. Mehrem, R.; Hohenegger, A.: A generalization for the infinite integral over three spherical Bessel functions (2010)
  10. Van Deun, Joris; Cools, Ronald: Integrating products of Bessel functions with an additional exponential or rational factor (2008)
  11. Van Deun, Joris; Cools, Ronald: A stable recurrence for the incomplete gamma function with imaginary second argument (2006)
  12. Van Deun, Joris; Cools, Ronald: A Matlab implementation of an algorithm for computing integrals of products of Bessel functions (2006)
  13. Van Deun, Joris; Cools, Ronald: Algorithm 858: Computing infinite range integrals of an arbitrary product of Bessel functions. (2006)