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.

Scalar case

Let f be a continuous function on the real line. Then the nth repeated integral of f based at a,
is given by single integration
A proof is given by induction. Since f 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. Firstly, using the Leibniz integral rule, note that
Then, applying the induction hypothesis,
This completes the proof.

Applications

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 ! as Γ. Differentiating a fractional number of times can be accomplished by fractional integration, then differentiating the result.