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 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:
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.
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 dimension 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.
- Gerald B. Folland, Advanced Calculus, p. 193, Prentice Hall (2002). ISBN 0-13-065265-2