Central Difference

The central difference for a function tabulated at equal intervals  is defined by

(1)

First and higher order central differences arranged so as to involve integer indices are then given by

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

(Abramowitz and Stegun 1972, p. 877).

Higher order differences may be computed for even and odd powers,

			(8)
			(9)

(Abramowitz and Stegun 1972, p. 877).

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Differences." §25.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877-878, 1972.

Jeffreys, H. and Jeffreys, B. S. "Central Differences Formula." §9.084 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 284-286, 1988.

Sheppard, W. F. "Central-Difference Formulæ." Proc. London Math. Soc. 31, 449-488, 1899.

Whittaker, E. T. and Robinson, G. "Central-Difference Formulae." Ch. 3 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 35-52, 1967.