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 (cf. Cauchy's formula).

Scalar case

Let ''f'' be a continuous function on the real line. Then the ''n''th repeated integral of ''f'' with basepoint ''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 \left(x-t\right)^n f(t)\,\mathrmt \right= \frac \int_a^x\left(x-t\right)^ 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 \frac \int_a^\left(\sigma_1-t\right)^ f(t)\,\mathrmt\,\mathrm\sigma_1 \\
&= \int_a^x \frac \left frac \int_a^ \left(\sigma_1-t\right)^n f(t)\,\mathrmt\right\,\mathrm\sigma_1 \\
&= \frac \int_a^x \left(x-t\right)^n f(t)\,\mathrmt.
\end

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 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.

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=Alan Beardon, url=http://nrich.maths.org/public/viewer.php?obj_id=1369, title=Fractional calculus II, publisher=University of Cambridge, year=2000

Augustin-Louis Cauchy
Integral calculus
Theorems in analysis