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
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. [1]
External links
- Darboux's formula at MathWorld