Darboux's formula is a theorem on the expansion of functions in 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 in a neighborhood of the line
segment joining 
to 
, and let 
be a polynomial of degree 
in 
. Then if 
, differentiation gives
 |
Since 
has degree 
, its 
th derivative is constant, so
. Integrating from 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) |
Substituting 
gives the finite Taylor series formula with an integral
remainder. The infinite Taylor series follows when
the remainder tends to 0 as 
, for example when 
has 
smaller than the radius
of convergence about 
(Whittaker and Watson 1990, p. 125).
