Darboux's Formula

Author

Eric W. Weisstein
February 17, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/DarbouxsFormula.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DarbouxsFormula.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Equation

f(z)==f((Subscript[z, 0]))+(1/((ϕ^n)(0)))((((-1))^n) ((∫_0^1)ϕ(t) Derivative[n+1][f](t (z-(Subscript[z, 0]))+(Subscript[z, 0]))\[DifferentialD]t) (((z-(Subscript[z, 0])))^(n+1))+(∑_(k=1)^n)(((-1))^(k-1)) (((z-(Subscript[z, 0])))^k) (Derivative[n-k][ϕ](1) Derivative[k][f](z)-Derivative[n-k][ϕ](0) Derivative[k][f]((Subscript[z, 0]))))/;ϕ(t)==(∑_(k=0)^n)(Subscript[a, k]) (t^k)