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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{align} & \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{align}

The formula can be proved by repeated integration by parts.

Special cases

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.

References

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.