In mathematical analysis, Darboux's formula is a formula introduced by for summing infinite series by using integrals or evaluating integrals using infinite series. It is a generalization to the complex plane of the EulerâÂÂMaclaurin 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.
If ÃÂ(t) is a polynomial of degree n and f an analytic function then
The formula can be proved by repeated integration by parts.
Taking φ to be a Bernoulli polynomial in Darboux's formula gives the EulerâÂÂMaclaurin summation formula. Taking àto be (t â 1)<sup>n</sup> gives the formula for a Taylor series.