Summation 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.
Statement.
If "φ"("t") is a polynomial of degree "n" and "f" an analytic function then
 formula_1
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.