We propose a general method for computing indefinite integrals of the form
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)|t–d|α e iωt, g(t)log|t–d| e iωt, g(t) t α J ν (ct).
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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
- Indefinite integration
- Oscillatory function
- Bessel function
- Linear difference equation
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