mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, Darboux's formula is a formula introduced by for

summing Summing (April 16, 1978 – October 10, 2008) was an American thoroughbred racehorse and sire. Background Summing was a bay horse bred in Kentucky by his owner Charles T. Wilson Jr. He was sired by Verbatim, a leading dirt performer whose wi ...

infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, ma ...

by using

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

s or evaluating integrals using infinite series. It is a generalization to the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...

of the Euler–Maclaurin summation formula, which is used for similar purposes and derived in a similar manner (by repeated

integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...

of a particular choice of

integrand In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

). Darboux's formula can also be used to derive the

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...

from

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

Statement

If ''φ''(''t'') is a polynomial of degree ''n'' and ''f'' an analytic function then :

, dt. \end

The formula can be proved by repeated

Special cases

Taking ''φ'' to be a

Bernoulli polynomial In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in ...

in Darboux's formula gives the Euler–Maclaurin summation formula. Taking ''φ'' to be (''t'' − 1)^''n'' gives the formula for a

References

*{{citation, last=Darboux, title=Sur les développements en série des fonctions d'une seule variable, year=1876, url=http://gallica.bnf.fr/ark:/12148/bpt6k16420b/f291, volume=3, issue=II, pages=291–312, journal= Journal de Mathématiques Pures et Appliquées * Whittaker, E. T. and Watson, G. N. "A Formula Due to Darboux." §7.1 in ''A Course in Modern Analysis'', 4th ed. Cambridge, England: Cambridge University Press, p. 125, 1990

External links

Darboux's formula
at

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Mathematical analysis Summability methods