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
![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.](/images/equations/DarbouxsFormula/NumberedEquation2.svg) |
The Taylor series follows by letting 
and letting 
(Whittaker and Watson 1990, p. 125).
