Cauchy formula for repeated integration

Let f be a continuous function on the real line. Then the nth repeated integral of f with base-point a,

$$f^\{(-n)\}(x)\; =\; \backslash int\_a^x\; \backslash int\_a^\{\backslash sigma\_1\}\; \backslash cdots\; \backslash int\_a^\{\backslash sigma\_\{n-1\}\}\; f(\backslash sigma\_\{n\})\; \backslash ,\; \backslash mathrm\{d\}\backslash sigma\_\{n\}\; \backslash cdots\; \backslash ,\; \backslash mathrm\{d\}\backslash sigma\_2\; \backslash ,\; \backslash mathrm\{d\}\backslash sigma\_1,$$

is given by single integration

$$f^\{(-n)\}(x)\; =\; \backslash frac\{1\}\{(n-1)!\}\; \backslash int\_a^x\backslash left(x-t\backslash right)^\{n-1\}\; f(t)\backslash ,\backslash mathrm\{d\}t.$$

A proof is given by induction. The base case with n = 1 is trivial, since it is equivalent to

f^{(-1)}(x) = \frac1{0!} \int_a^x {(x - t)^0} f(t)\,\mathrm{d}t = \int_a^x f(t)\,\mathrm{d}t.

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

\frac{\mathrm{d}}{\mathrm{d}x} \left[ \frac{1}{n!} \int_a^x (x - t)^n f(t)\,\mathrm{d}t \right] =

\frac{1}{(n - 1)!} \int_a^x (x - t)^{n-1} f(t)\,\mathrm{d}t.

Then, applying the induction hypothesis,

$$\backslash begin\{align\}$$

f^{-(n+1)}(x) &= \int_a^x \int_a^{\sigma_1} \cdots \int_a^{\sigma_n} f(\sigma_{n+1}) \,\mathrm{d}\sigma_{n+1} \cdots \,\mathrm{d}\sigma_2 \,\mathrm{d}\sigma_1 \\

&= \int_a^x \left[\int_a^{\sigma_1} \cdots \int_a^{\sigma_n} f(\sigma_{n+1}) \,\mathrm{d}\sigma_{n+1} \cdots \,\mathrm{d}\sigma_2 \right] \,\mathrm{d}\sigma_1.

\end{align}

Note that the term within square bracket has n-times successive integration, and upper limit of outermost integral inside the square bracket is $\backslash sigma\_1$. Thus, comparing with the case for n = n and replacing $x,\; \backslash sigma\_1,\; \backslash cdots,\; \backslash sigma\_n$ of the formula at induction step n = n with $\backslash sigma\_1,\; \backslash sigma\_2,\; \backslash cdots,\; \backslash sigma\_\{n+1\}$ respectively leads to

\int_a^{\sigma_1} \cdots \int_a^{\sigma_n} f(\sigma_{n+1}) \,\mathrm{d}\sigma_{n+1} \cdots \,\mathrm{d}\sigma_2 = \frac{1}{(n - 1)!} \int_a^{\sigma_1} (\sigma_1 - t)^{n-1} f(t)\,\mathrm{d}t.

Putting this expression inside the square bracket results in

$$\backslash begin\{align\}$$

&= \int_a^x \frac{1}{(n - 1)!} \int_a^{\sigma_1} (\sigma_1 - t)^{n-1} f(t)\,\mathrm{d}t\,\mathrm{d}\sigma_1 \\

&= \int_a^x \frac{\mathrm{d}}{\mathrm{d}\sigma_1} \left[\frac{1}{n!} \int_a^{\sigma_1} (\sigma_1 - t)^n f(t)\,\mathrm{d}t\right] \,\mathrm{d}\sigma_1 \\

&= \frac{1}{n!} \int_a^x (x - t)^n f(t)\,\mathrm{d}t.

\end{align}

• 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.

The Cauchy formula is generalized to non-integer parameters by the Riemann–Liouville integral, where $n\; \backslash in\; \backslash Z\_\{\backslash geq\; 0\}$ is replaced by $\backslash alpha\; \backslash in\; \backslash Complex,\backslash \; \backslash Re(\backslash alpha)\; >\; 0$, and the factorial is replaced by the gamma function. The two formulas agree when $\backslash alpha\; \backslash in\; \backslash Z\_\{\backslash geq\; 0\}$.

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.

• Augustin-Louis Cauchy: [https://gallica.bnf.fr/ark:/12148/bpt6k62404287/f150.item 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). {{ISBN|0-13-065265-2}}

• {{cite web|author=Alan Beardon| url=http://nrich.maths.org/public/viewer.php?obj_id=1369| title=Fractional calculus II| publisher=University of Cambridge| year=2000}}

• {{cite web|author=Maurice Mischler| url=https://sites.google.com/site/mathmontmus/accueil/pages-math%C3%A9matiques-dures/int%C3%A9grales-ni%C3%A8mes-et-polyn%C3%B4mes-sympas| title= About some repeated integrals and associated polynomials| year=2023}}

Category:Augustin-Louis Cauchy

Category:Integral calculus

Category:Theorems in mathematical analysis