Darboux's formula

Short description: 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

$\begin{array}{rl}& \sum _{m=0}^n(-1{)}^m(z-a{)}^m[{\phi }^{(n-m)}(1)f^{(m)}(z)-{\phi }^{(n-m)}(0)f^{(m)}(a)]\\ =& (-1{)}^n(z-a{)}^{n+1}\underset{0}{\overset{1}{\int }}\phi (t)f^{(n+1)}[a+t(z-a)]dt.\end{array}$

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