Cauchy Formula For Repeated Integration
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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 nth repeated integral of f based at a,

,

is given by single integration

.

A proof is given by induction. Since f 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.

Generalizations and Applications

The Cauchy formula is generalized to non-integer parameters by the Riemann-Liouville integral, where is replaced by , and the factorial is replaced by the gamma function. The two formulas agree when .

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

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

External links


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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