# Indefinite integration of oscillatory functions

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

top

## Abstract

top
A simple and fast algorithm is presented for evaluating the indefinite integral of an oscillatory function ${\int }_{x}^{y}if\left(t\right){e}^{i\omega t}dt$, -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.

## How to cite

top

Keller, 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. [1] W. Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24-82. Zbl0168.15004
2. [2] W. M. Gentleman, Implementing Clenshaw-Curtis quadrature. II. Computing the cosine transformation, Comm. ACM 15 (1972), 343-346. Zbl0234.65024
3. [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. [4] T. Hasegawa and T. Torii, Application of a modified FFT to product type integration, ibid. 38 (1991), 157-168. Zbl0754.65023
5. [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. [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. [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. [8] S. Paszkowski, Numerical Applications of Chebyshev Polynomials and Chebyshev Series, PWN, Warszawa, 1975 (in Polish). Zbl0423.65012

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.