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Darboux's Formula

Darboux's formula is a theorem on the expansion of functions in infinite series and essentially consists of integration by parts on a specific integrand product of functions. Taylor series may be obtained as a special case of the formula, which may be stated as follows.

Let be analytic at all points of the line joining to , and let be any polynomial of degree in . Then if , differentiation gives

d/(dt)sum_(m=1)^infty(-1)^m(z-a)^mphi^((n-m))(t)f^((m))(a+t(z-a))
=-(z-a)phi^((n))(t)f^'(a+t(z-a))+(-1)^n(z-a)^(n+1)phi(t)f^((n+1))(a+t(z-a)).

But , so integrating over the interval 0 to 1 gives

phi^((n))(0)[f(z)-f(a)]=sum_(m=1)^n(-1)^(m-1)(z-a)^m[phi^((n-m))(1)f^((m))(z)-phi^((n-m))(0)f^((m))(a)]+(-1)^n(z-a)^(n+1)int_0^1phi(t)f^((n+1))(a+t(z-a))dt.

The Taylor series follows by letting and letting (Whittaker and Watson 1990, p. 125).

REFERENCES:

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.

CITE THIS AS:

Weisstein, Eric W. "Darboux's Formula." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DarbouxsFormula.html