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Filon's Integration Formula

A formula for Numerical Integration,

$\int_{x_0}^{x_n} f(x)\cos(tx)\,dx =h\{\alpha(th)[f_{2n}\sin(tx_{2n})-f_0\sin(tx_0)]$
$ +\beta(th)C_{2n}+\gamma(th)C_{2n-1}+{\textstyle{2\over 45}}th^4 S_{2n-1}'\}-R_n,\quad$ (1)

$\displaystyle C_{2n}$ $\textstyle =$ $\displaystyle \sum_{i=0}^n f_{2i}\cos(tx_{2i})-{\textstyle{1\over 2}}[f_{2n}\cos(tx_{2n})+f_0\cos(tx_0)]$  
$\displaystyle C_{2n-1}$ $\textstyle =$ $\displaystyle \sum_{i=1}^n f_{2i-1}\cos(tx_{2i-1})$ (3)
$\displaystyle S_{2n-1}'$ $\textstyle =$ $\displaystyle \sum_{i=1}^n f^{(3)}_{2i-1} \sin(tx_{2i-1})$ (4)
$\displaystyle \alpha(\theta)$ $\textstyle =$ $\displaystyle {1\over\theta}+{\sin(2\theta)\over 2\theta^2}-{2\sin^2\theta\over\theta^3}$ (5)
$\displaystyle \beta(\theta)$ $\textstyle =$ $\displaystyle 2\left[{{1+\cos^2\theta\over\theta^2}-{\sin(2\theta)\over\theta^3}}\right]$ (6)
$\displaystyle \gamma(\theta)$ $\textstyle =$ $\displaystyle 4\left({{\sin\theta\over\theta^3}-{\cos\theta\over\theta^2}}\right),$ (7)

and the remainder term is
R_n={\textstyle{1\over 90}} nh^5f^{(4)}(\xi)+{\mathcal O}(th^7).
\end{displaymath} (8)


Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 890-891, 1972.

Tukey, J. W. In On Numerical Approximation: Proceedings of a Symposium Conducted by the Mathematics Research Center, United States Army, at the University of Wisconsin, Madison, April 21-23, 1958 (Ed. R. E. Langer). Madison, WI: University of Wisconsin Press, p. 400, 1959.

© 1996-9 Eric W. Weisstein