Darboux's formula
# 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.

## Statement

If φ(t) is a polynomial of degree n and f an analytic function then

    ${\begin{aligned}&\sum _{m=0}^{n}(-1)^{m}(z-a)^{m}\left[\varphi ^{(n-m)}(1)f^{(m)}(z)-\varphi ^{(n-m)}(0)f^{(m)}(a)\right]\\=&(-1)^{n}(z-a)^{n+1}\int _{0}^{1}\varphi (t)f^{(n+1)}\left[a+t(z-a)\right]\,dt.\end{aligned}}$

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.

