Darboux's formula

Not to be confused with Christoffel–Darboux 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

$\begin{align} & \sum_{m=0}^n (-1)^{m}(z-a)^m\left(\phi^{(n-m)}(1)f^{(m)}(z) -\phi^{(n-m)}(0)f^{(m)}(a)\right) \\ & =(-1)^n(z-a)^{n+1}\int_0^1\phi(t)f^{(n+1)}(a+t(z-a))\, dt \end{align}$

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, http://gallica.bnf.fr/ark:/12148/bpt6k16420b/f291 
• 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