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Everett's Formula

f_p=(1-p)f_0+pf_1+E_2delta_0^2+F_2delta_1^2+E_4delta_0^4+F_4delta_1^4+E_6delta_0^6+F_6delta_1^6+...,
(1)

for p in [0,1], where delta is the central difference and

E_(2n)=G_(2n)-G_(2n+1)
(2)
=B_(2n)-B_(2n+1)
(3)
F_(2n)=G_(2n+1)
(4)
=B_(2n)+B_(2n+1),
(5)

where G_k are the coefficients from Gauss's backward formula and Gauss's forward formula and B_k are the coefficients from Bessel's finite difference formula. The E_ks and F_ks also satisfy

E_(2n)(p)=F_(2n)(q)
(6)
F_(2n)(p)=E_(2n)(q),
(7)

for

q=1-p.
(8)

See also

Bessel's Finite Difference Formula

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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. 880-881, 1972.Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., pp. 92-93, 1990.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 433, 1987.Whittaker, E. T. and Robinson, G. "The Laplace-Everett Formula." §25 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 40-41, 1967.

Referenced on Wolfram|Alpha

Everett's Formula

Cite this as:

Weisstein, Eric W. "Everett's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EverettsFormula.html

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