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).

Scalar case[edit]

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. Since ƒ is continuous, the base case follows from the Fundamental theorem of calculus:

;

where

.

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

.

Then, applying the induction hypothesis,

This completes the proof.

Applications[edit]

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[edit]

  • Gerald B. Folland, Advanced Calculus, p. 193, Prentice Hall (2002). ISBN 0-13-065265-2

External links[edit]