Cauchy formula for repeated integration
![]() | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (May 2024) |
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
[edit]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
[edit]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, the term within square bracket has n-times successive integration, and upper limit of outermost integral inside the square bracket is . Thus, comparing with the case for n=n, and replacing of the formula at induction step n=n with respectively to obtain
Putting this expression inside the square bracket results in
- It has been shown that this statement holds true for the base case .
- If the statement is true for , then it has been shown that the statement holds true for .
- Thus this statement has been proven true for all positive integers.
This completes the proof.
Generalizations and applications
[edit]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.
References
[edit]- Augustin-Louis Cauchy: Trente-Cinquième Leçon. In: Résumé des leçons données à l’Ecole royale polytechnique sur le calcul infinitésimal. Imprimerie Royale, Paris 1823. Reprint: Œuvres complètes II(4), Gauthier-Villars, Paris, pp. 5–261.
- Gerald B. Folland, Advanced Calculus, p. 193, Prentice Hall (2002). ISBN 0-13-065265-2
External links
[edit]- Alan Beardon (2000). "Fractional calculus II". University of Cambridge.
- Maurice Mischler (2023). "About some repeated integrals and associated polynomials".