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Cauchy formula for repeated integration

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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).

Contents

Scalar case

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

f ( − n ) ( x ) = ∫ a x ∫ a σ 1 ⋯ ∫ a σ n − 1 f ( σ n ) d σ n ⋯ d σ 2 d σ 1 ,

is given by single integration

f ( − n ) ( x ) = 1 ( n − 1 ) ! ∫ a x ( x − t ) n − 1 f ( t ) d t .

A proof is given by induction. Since ƒ is continuous, the base case follows from the Fundamental theorem of calculus:

d d x f ( − 1 ) ( x ) = d d x ∫ a x f ( t ) d t = f ( x ) ;

where

f ( − 1 ) ( a ) = ∫ a a f ( t ) d t = 0 .

Now, suppose this is true for n, and let us prove it for n+1. Apply the induction hypothesis and switching the order of integration,

f − ( n + 1 ) ( x ) = ∫ a x ∫ a σ 1 ⋯ ∫ a σ n f ( σ n + 1 ) d σ n + 1 ⋯ d σ 2 d σ 1 = 1 ( n − 1 ) ! ∫ a x ∫ a σ 1 ( σ 1 − t ) n − 1 f ( t ) d t d σ 1 = 1 ( n − 1 ) ! ∫ a x ∫ t x ( σ 1 − t ) n − 1 f ( t ) d σ 1 d t = 1 n ! ∫ a x ( x − t ) n f ( t ) d t

This completes the proof.

Applications

In fractional calculus, 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). Differentiating a fractional number of times can be accomplished by fractional integration, then differentiating the result.

References

Cauchy formula for repeated integration Wikipedia
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