Darboux's formula

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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{aligned}&\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)}}\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

External links

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