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Darboux's formula

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Summation 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 from calculus.

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\\={}&(-1)^{n}(z-a)^{n+1}\int _{0}^{1}\varphi (t)f^{(n+1)}\left\,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) gives the formula for a Taylor series.

References

External links

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