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Gregory's Formula

Gregory's formula is a formula that allows a definite integral of a function to be expressed by its sum and differences, or its sum by its integral and difference (Jordan 1965, p. 284). It is given by the equation

int_0^yp(u)du=sum_(k>=0)(<(e^(yt)-1)^k|p(x)>)/(k!)(e^t-1)^kp(x),

discovered by Gregory in 1670 and reported to be the earliest formula in numerical integration (Jordan 1965, Roman 1984).

See also

Gregory Series, Leibniz Series, Machin's Formula, Machin-Like Formulas, Numerical Integration, Pi Formulas

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References

Jordan, C. "Gregory's Summation Formula." §99 in Calculus of Finite Differences, 3rd ed. New York: Chelsea, pp. 284-287, 1965.Roman, S. The Umbral Calculus. New York: Academic Press, p. 59, 1984.

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Gregory's Formula

Cite this as:

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

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