Cauchy formula for repeated integration explained

The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress n antiderivatives 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 with base-point a,$$f^(x)\; =\; \backslash int\_a^x\; \backslash int\_a^\; \backslash cdots\; \backslash int\_a^\; f(\backslash sigma\_)\; \backslash ,\; \backslash mathrm\backslash sigma\_\; \backslash cdots\; \backslash ,\; \backslash mathrm\backslash sigma\_2\; \backslash ,\; \backslash mathrm\backslash sigma\_1,$$is given by single integration$$f^(x)\; =\; \backslash frac\; \backslash int\_a^x\backslash left(x-t\backslash right)^\; f(t)\backslash ,\backslash mathrmt.$$

Proof

A proof is given by induction. The base case with n = 1 is trivial, since it is equivalent to$$f^(x)\; =\; \backslash frac1\; \backslash int\_a^x\; f(t)\backslash ,\backslash mathrmt\; =\; \backslash int\_a^x\; f(t)\backslash ,\backslash mathrmt.$$

Now, suppose this is true for n, and let us prove it for n + 1. Firstly, using the Leibniz integral rule, note that$$\backslash frac\; \backslash left[\backslash frac\{1\}\{n!\}\; \backslash int\_a^x\; (x\; -\; t)^n\; f(t)\backslash ,\backslash mathrm\{d\}t\; \backslash right]\; =\; \backslash frac\; \backslash int\_a^x\; (x\; -\; t)^\; f(t)\backslash ,\backslash mathrmt.$$Then, applying the induction hypothesis,Note that the term within square bracket has n-times successive integration, and upper limit of outermost integral inside the square bracket is

\sigma _{1} . Thus, comparing with the case for n = n and replacing x, \sigma _{1,} … , \sigma _{n} of the formula at induction step n = n with \sigma _{1,} \sigma _{2,} … , \sigma _{n+1} respectively leads to$$\backslash int\_a^\; \backslash cdots\; \backslash int\_a^\; f(\backslash sigma\_)\; \backslash ,\backslash mathrm\backslash sigma\_\; \backslash cdots\; \backslash ,\backslash mathrm\backslash sigma\_2\; =\; \backslash frac\; \backslash int\_a^\; (\backslash sigma\_1\; -\; t)^\; f(t)\backslash ,\backslash mathrmt.$$Putting this expression inside the square bracket results in

• It has been shown that this statement holds true for the base case

n = 1 .

• If the statement is true for

n = k , then it has been shown that the statement holds true for n = k + 1 .

• Thus this statement has been proven true for all positive integers.

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 _{\geq} is replaced by \alpha \in \Complex, \Re(\alpha) > 0 , and the factorial is replaced by the gamma function. The two formulas agree when \alpha \in \Z _{\geq} .

Both the Cauchy formula and the Riemann–Liouville integral are generalized to arbitrary dimensions 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

• 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

• Web site: Alan Beardon. Fractional calculus II. University of Cambridge. 2000.
• Web site: Maurice Mischler. About some repeated integrals and associated polynomials. 2023.