Darboux's formula

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[citation needed][dubious – discuss].

Statement

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

∑ m = 0 n ( − 1 ) m ( z − a ) m [ φ ( n − m ) ( 1 ) f ( m ) ( z ) − φ ( n − m ) ( 0 ) f ( m ) ( a ) ] = ( − 1 ) n ( z − a ) n + 1 ∫ 0 1 φ ( t ) f ( n + 1 ) [ a + t ( z − a ) ] d t . {\displaystyle {\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.

References

  • Darboux (1876), "Sur les développements en série des fonctions d'une seule variable", Journal de Mathématiques Pures et Appliquées, 3 (II): 291–312
  • 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]
  • Darboux's formula at MathWorld