# Indefinite integration of oscillatory functions

Applicationes Mathematicae (1998)

- Volume: 25, Issue: 3, page 301-311
- ISSN: 1233-7234

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topKeller, Paweł. "Indefinite integration of oscillatory functions." Applicationes Mathematicae 25.3 (1998): 301-311. <http://eudml.org/doc/219205>.

@article{Keller1998,

abstract = {A simple and fast algorithm is presented for evaluating the indefinite integral of an oscillatory function $\int _x^yi f(t) e^\{iωt\} dt$, -1 ≤ x < y ≤ 1, ω ≠ 0, where the Chebyshev series expansion of the function f is known. The final solution, expressed as a finite Chebyshev series, is obtained by solving a second-order linear difference equation. Because of the nature of the equation special algorithms have to be used to find a satisfactory approximation to the integral.},

author = {Keller, Paweł},

journal = {Applicationes Mathematicae},

keywords = {indefinite integration; second-order linear difference equation; oscillatory function; trigonometric approximation; algorithm; Chebyshev series},

language = {eng},

number = {3},

pages = {301-311},

title = {Indefinite integration of oscillatory functions},

url = {http://eudml.org/doc/219205},

volume = {25},

year = {1998},

}

TY - JOUR

AU - Keller, Paweł

TI - Indefinite integration of oscillatory functions

JO - Applicationes Mathematicae

PY - 1998

VL - 25

IS - 3

SP - 301

EP - 311

AB - A simple and fast algorithm is presented for evaluating the indefinite integral of an oscillatory function $\int _x^yi f(t) e^{iωt} dt$, -1 ≤ x < y ≤ 1, ω ≠ 0, where the Chebyshev series expansion of the function f is known. The final solution, expressed as a finite Chebyshev series, is obtained by solving a second-order linear difference equation. Because of the nature of the equation special algorithms have to be used to find a satisfactory approximation to the integral.

LA - eng

KW - indefinite integration; second-order linear difference equation; oscillatory function; trigonometric approximation; algorithm; Chebyshev series

UR - http://eudml.org/doc/219205

ER -

## References

top- [1] W. Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24-82. Zbl0168.15004
- [2] W. M. Gentleman, Implementing Clenshaw-Curtis quadrature. II. Computing the cosine transformation, Comm. ACM 15 (1972), 343-346. Zbl0234.65024
- [3] T. Hasegawa and T. Torii, Indefinite integration of oscillatory functions by the Chebyshev series expansion, J. Comput. Appl. Math. 17 (1987), 21-29. Zbl0613.65145
- [4] T. Hasegawa and T. Torii, Application of a modified FFT to product type integration, ibid. 38 (1991), 157-168. Zbl0754.65023
- [5] T. Hasegawa and T. Torii, An algorithm for nondominant solutions of linear second-order inhomogeneous difference equations, Math. Comp. 64 (1995), 1199-1204. Zbl0831.65140
- [6] T. Hasegawa, T. Torii and H. Sugiura, An algorithm based on the FFT for a generalized Chebyshev interpolation, ibid. 54 (1990), 195-210. Zbl0685.65003
- [7] F. W. J. Olver, Numerical solution of second-order linear difference equations, J. Res. Nat. Bur. Standards 71 (B) (1967), 111-129. Zbl0171.36601
- [8] S. Paszkowski, Numerical Applications of Chebyshev Polynomials and Chebyshev Series, PWN, Warszawa, 1975 (in Polish). Zbl0423.65012

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