Cauchy formula for repeated integration

The Cauchy formula for repeated integration, named after Augustin Louis Cauchy, allows one to compress n antidifferentiations of a function into a single integral (cf. Cauchy's formula).

Let ƒ be a continuous function on the real line. Then the nth repeated integral of ƒ based at a,

is given by single integration

A proof is given by induction. Since ƒ is continuous, the base case follows from the Fundamental theorem of calculus:

where

Now, suppose this is true for n, and let us prove it for n+1. Apply the induction hypothesis and switching the order of integration,

This completes the proof.

In fractional calculus, this formula can be used to construct a notion of differintegral, allowing one to differentiate or integrate a fractional number of times. Integrating a fractional number of times with this formula is straightforward; one can use fractional n by interpreting (n-1)! as Γ(n) (see Gamma function). Differentiating a fractional number of times can be accomplished by fractional integration, then differentiating the result.

...
Wikipedia