The Cauchy formula for repeated integration, named after Augustin Louis Cauchy

Augustin Louis Cauchy

Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...

, allows one to compress n antidifferentiations of a function into a single integral (cf. Cauchy's formula).

Scalar case

Let ƒ be a continuous function on the real line. Then the nth repeated integral of ƒ based at a,
,

is given by single integration
.

A proof is given by induction

Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

. Since ƒ is continuous, the base case is given by noting that


and that
.

Now, suppose this is true for n; to prove it for n+1, we apply the chain rule. Consider the function
;

then we have


and, applying differentiation under the integral sign, we have that
.

Thus,
.

Furthermore,
.

Hence, ƒ^(-n) is an nth antiderivative of ƒ, and ƒ^(-k)(a)=0 for all k from 1 to n, showing that ƒ^(-n)(x) is equal to the original repeated integral.

Applications

In fractional calculus

Fractional calculus

Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.and the integration operator J...

, this formula can be used to construct a notion of differintegral, allowing one to differentiate or integrate a fractional number of times. Integrating a fractional number of times with this formula is straightforward; one can use fractional n by interpreting (n-1)! as Γ(n) (see Gamma function

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

).