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 from calculus[citation needed][dubious ].
Statement
If φ(t) is a polynomial of degree n and f an analytic function then
- [math]\displaystyle{ \begin{align} & \sum_{m=0}^n (-1)^m (z - a)^m \left[\varphi^{(n - m)}(1)f^{(m)}(z) - \varphi^{(n - m)}(0)f^{(m)}(a)\right] \\ = {} & (-1)^n(z - a)^{n + 1}\int_0^1\varphi(t)f^{(n+1)}\left[a + t(z - a)\right]\, dt. \end{align} }[/math]
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, http://gallica.bnf.fr/ark:/12148/bpt6k16420b/f291
- 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. [1]
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