Darboux's formula

In mathematical analysis, Darboux's formula is a formula introduced by Gaston Darboux (1876) for summing infinite series by using integrals or evaluating integrals using infinite series. It is a generalization to the complex plane of the Euler–Maclaurin summation formula, which is used for similar purposes and derived in a similar manner (by repeated integration by parts of a particular choice of integrand). Darboux's formula can also be used to derive the Taylor series of the calculus.

Statement

If φ(t) is a polynomial of degree n and f an analytic function then

{\begin{aligned}&\sum _{{m=0}}^{n}(-1)^{m}(z-a)^{m}\left[\phi ^{{(n-m)}}(1)f^{{(m)}}(z)-\phi ^{{(n-m)}}(0)f^{{(m)}}(a)\right]\\{}=&(-1)^{n}(z-a)^{{n+1}}\int _{0}^{1}\phi (t)f^{{(n+1)}}\left[a+t(z-a)\right]\,dt.\end{aligned}}

The formula can be proved by repeated integration by parts.

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

Darboux (1876), "Sur les développements en série des fonctions d'une seule variable", Journal de Mathématiques Pures et Appliquées, 3 (II): 291–312
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.

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