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

But , so integrating over the interval 0 to 1 gives

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