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

Scalar case

Let f be a continuous function on the real line. Then the nth repeated integral of f with base-point a, is given by single integration

Proof

A proof is given by induction. The base case with n = 1 is trivial, since it is equivalent to 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, Note that the term within square bracket has n-times successive integration, and upper limit of outermost integral inside the square bracket is \sigma_1. Thus, comparing with the case for n = n and replacing of the formula at induction step n = n with respectively leads to Putting this expression inside the square bracket results in This completes the proof.

Generalizations and applications

The Cauchy formula is generalized to non-integer parameters by the Riemann–Liouville integral, where is replaced by, and the factorial is replaced by the gamma function. The two formulas agree when. Both the Cauchy formula and the Riemann–Liouville integral are generalized to arbitrary dimensions by the Riesz potential. In fractional calculus, these formulae can be used to construct a differintegral, allowing one to differentiate or integrate a fractional number of times. Differentiating a fractional number of times can be accomplished by fractional integration, then differentiating the result.