A method for indefinite integration of oscillatory and singular functions

Abstract

We propose a general method for computing indefinite integrals of the form

$$I(y) = {\int_0^y {g(t)k} }(t)dt\;\;\;\;(0 \leqslant y \leqslant y_{{{\text{max}}}} ),$$

where g is a smooth function, and k is a function that contains a singular factor or is rapidly oscillatory. The only assumption on k is that it satisfies a linear differential equation with polynomial coefficients. The approximate value of the integral is given in terms of Chebyshev coefficients of functions that form a solution of a certain system of differential equations. As an illustration, we present effective algorithms for computing indefinite integrals of the functions g(t)|td|α e iωt, g(t)log|td| e iωt, g(t) t α J ν (ct).

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Correspondence to Paweł Keller.

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Keller, P. A method for indefinite integration of oscillatory and singular functions. Numer Algor 46, 219–251 (2007). https://doi.org/10.1007/s11075-007-9134-y

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Keywords

  • Indefinite integration
  • Oscillatory function
  • Bessel function
  • Linear difference equation

Mathematics Subject Classifications (2000)

  • 65D30
  • 65Q05