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 Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...

. 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 In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form \int_^ f(x,t)\,dt, where -\infty < a(x), b(x) < \infty and the integral are

, note that

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

Then, applying the induction hypothesis,

\,\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 In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

. 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 mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to ...

. In

fractional calculus Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D :D f(x) = \frac f(x)\,, and of the integration ...

, these formulae can be used to construct a

differintegral In fractional calculus, an area of mathematical analysis, the differintegral (sometime also called the derivigral) is a combined differentiation/ integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted by ...

, 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