Darboux's formula should not be confused with Christoffel–Darboux formula.
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 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.
If φ(t) is a polynomial of degree n and f an analytic function then
\begin{align} &
| n | |
| \sum | |
| m=0 |
(-1)m(z-a)m\left[\varphi(n(1)f(m)(z)-\varphi(n(0)f(m)(a)\right]\\ ={}&(-1)n(z-a)n
| 1\varphi(t)f | |
| \int | |
| 0 |
(n+1)\left[a+t(z-a)\right]dt. \end{align}
The formula can be proved by repeated integration by parts.
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.