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Filon's Integration Formula

A formula for numerical integration,

int_(x_0)^(x_(2n))f(x)cos(tx)dx=h{alpha(th)[f_(2n)sin(tx_(2n))-f_0sin(tx_0)] +beta(th)C_(2n)+gamma(th)C_(2n-1)+2/(45)th^4S_(2n-1)^'}-R_n,
(1)

where

C_(2n)=sum_(i=0)^(n)f_(2i)cos(tx_(2i))-1/2[f_(2n)cos(tx_(2n))+f_0cos(tx_0)]
(2)
C_(2n-1)=sum_(i=1)^(n)f_(2i-1)cos(tx_(2i-1))
(3)
S_(2n-1)^'=sum_(i=1)^(n)f_(2i-1)^((3))sin(tx_(2i-1))
(4)
alpha(theta)=1/theta+(sin(2theta))/(2theta^2)-(2sin^2theta)/(theta^3)
(5)
beta(theta)=2[(1+cos^2theta)/(theta^2)-(sin(2theta))/(theta^3)]
(6)
gamma(theta)=4((sintheta)/(theta^3)-(costheta)/(theta^2)),
(7)

and the remainder term is

R_n=1/(90)nh^5f^((4))(xi)+O(th^7).
(8)

See also

Numerical Integration

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References

Abramowitz, M. and Stegun, I. 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.

Referenced on Wolfram|Alpha

Filon's Integration Formula

Cite this as:

Weisstein, Eric W. "Filon's Integration Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FilonsIntegrationFormula.html

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