Cauchy Formula For Repeated Integration

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). For non-integer ''n'' it yields the definition of fractional integrals and (with ''n'' < 0) fractional derivatives.

Scalar case

Let ''f'' be a continuous function on the real line. Then the ''n''th repeated integral of ''f'' with base-point ''a'',
$f^(x) = \int_a^x \int_a^ \cdots \int_a^ f(\sigma_) \, \mathrm\sigma_ \cdots \, \mathrm\sigma_2 \, \mathrm\sigma_1,$
is given by single integration
$f^(x) = \frac \int_a^x\left(x-t\right)^ f(t)\,\mathrmt.$

Proof

A proof is given by induction. The base case with ''n'' = 1 is trivial, since it is equivalent to
$f^(x) = \frac1 \int_a^x f(t)\,\mathrmt = \int_a^x f(t)\,\mathrmt.$

Now, suppose this is true for ''n'', and let us prove it for ''n'' + 1. Firstly, using the Leibniz integral rule, note that

\frac \left \frac \int_a^x (x - t)^n f(t)\,\mathrmt \right=
\frac \int_a^x (x - t)^ f(t)\,\mathrmt.

Then, applying the induction hypothesis,
\begin
f^(x) &= \int_a^x \int_a^ \cdots \int_a^ f(\sigma_) \,\mathrm\sigma_ \cdots \,\mathrm\sigma_2 \,\mathrm\sigma_1 \\
&= \int_a^x \left int_a^ \cdots \int_a^ f(\sigma_) \,\mathrm\sigma_ \cdots \,\mathrm\sigma_2 \right\,\mathrm\sigma_1.
\end
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 $x, \sigma_1, \cdots, \sigma_n$ of the formula at induction step ''n'' = ''n'' with $\sigma_1, \sigma_2, \cdots, \sigma_$ respectively leads to
$\int_a^ \cdots \int_a^ f(\sigma_) \,\mathrm\sigma_ \cdots \,\mathrm\sigma_2 = \frac \int_a^ (\sigma_1 - t)^ f(t)\,\mathrmt.$
Putting this expression inside the square bracket results in
\begin
&= \int_a^x \frac \int_a^ (\sigma_1 - t)^ f(t)\,\mathrmt\,\mathrm\sigma_1 \\
&= \int_a^x \frac \left frac \int_a^ (\sigma_1 - t)^n f(t)\,\mathrmt\right\,\mathrm\sigma_1 \\
&= \frac \int_a^x (x - t)^n f(t)\,\mathrmt.
\end

* It has been shown that this statement holds true for the base case $n = 1$.
* If the statement is true for $n = k$, then it has been shown that the statement holds true for $n = k + 1$.
* Thus this statement has been proven true for all positive integers.

This completes the proof.

Generalizations and applications

The Cauchy formula is generalized to non-integer parameters by the Riemann–Liouville integral, where $n \in \Z_$ is replaced by $\alpha \in \Complex,\ \Re(\alpha) > 0$, and the factorial is replaced by the gamma function. The two formulas agree when $\alpha \in \Z_$.

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

* 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).

External links

*

*{{cite web, author=Maurice Mischler, url=https://sites.google.com/site/mathmontmus/accueil/pages-math%C3%A9matiques-dures/int%C3%A9grales-ni%C3%A8mes-et-polyn%C3%B4mes-sympas, title= About some repeated integrals and associated polynomials, year=2023

Augustin-Louis Cauchy
Integral calculus
Theorems in mathematical analysis