Cauchy formula for repeated integration

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).

Let f be a continuous function on the real line. Then the nth repeated integral of f with base-point a, $${\displaystyle f^{(-n)}(x)=\int _{a}^{x}\int _{a}^{\sigma _{1}}\cdots \int _{a}^{\sigma _{n-1}}f(\sigma _{n})\,\mathrm {d} \sigma _{n}\cdots \,\mathrm {d} \sigma _{2}\,\mathrm {d} \sigma _{1},}$$ is given by single integration $${\displaystyle f^{(-n)}(x)={\frac {1}{(n-1)!}}\int _{a}^{x}\left(x-t\right)^{n-1}f(t)\,\mathrm {d} t.}$$

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

$${\displaystyle f^{(-1)}(x)={\frac {1}{0!}}\int _{a}^{x}{(x-t)^{0}}f(t)\,\mathrm {d} t=\int _{a}^{x}f(t)\,\mathrm {d} t}$$ Now, suppose this is true for n, and let us prove it for n+1. Firstly, using the Leibniz integral rule, note that

$${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left[{\frac {1}{n!}}\int _{a}^{x}\left(x-t\right)^{n}f(t)\,\mathrm {d} t\right]={\frac {1}{(n-1)!}}\int _{a}^{x}\left(x-t\right)^{n-1}f(t)\,\mathrm {d} t.}$$

Then, applying the induction hypothesis,

$${\displaystyle {\begin{aligned}f^{-(n+1)}(x)&=\int _{a}^{x}\int _{a}^{\sigma _{1}}\cdots \int _{a}^{\sigma _{n}}f(\sigma _{n+1})\,\mathrm {d} \sigma _{n+1}\cdots \,\mathrm {d} \sigma _{2}\,\mathrm {d} \sigma _{1}\\&=\int _{a}^{x}\left[\int _{a}^{\sigma _{1}}\cdots \int _{a}^{\sigma _{n}}f(\sigma _{n+1})\,\mathrm {d} \sigma _{n+1}\cdots \,\mathrm {d} \sigma _{2}\right]\,\mathrm {d} \sigma _{1}\\\end{aligned}}}$$

Note, the term within square bracket has n-times successive integration, and upper limit of outermost integral inside the square bracket is ${\displaystyle \sigma _{1}}$. Thus, comparing with the case for n=n, and replacing ${\displaystyle x,\sigma _{1},\cdots ,\sigma _{n}}$ of the formula at induction step n=n with ${\displaystyle \sigma _{1},\sigma _{2},\cdots ,\sigma _{n+1}}$ respectively to obtain

$${\displaystyle {\begin{aligned}\int _{a}^{\sigma _{1}}\cdots \int _{a}^{\sigma _{n}}f(\sigma _{n+1})\,\mathrm {d} \sigma _{n+1}\cdots \,\mathrm {d} \sigma _{2}&={\frac {1}{(n-1)!}}\int _{a}^{\sigma _{1}}\left(\sigma _{1}-t\right)^{n-1}f(t)\,\mathrm {d} t\\\end{aligned}}}$$

Putting this expression inside the square bracket results in

$${\displaystyle {\begin{aligned}&=\int _{a}^{x}{\frac {1}{(n-1)!}}\int _{a}^{\sigma _{1}}\left(\sigma _{1}-t\right)^{n-1}f(t)\,\mathrm {d} t\,\mathrm {d} \sigma _{1}\\&=\int _{a}^{x}{\frac {\mathrm {d} }{\mathrm {d} \sigma _{1}}}\left[{\frac {1}{n!}}\int _{a}^{\sigma _{1}}\left(\sigma _{1}-t\right)^{n}f(t)\,\mathrm {d} t\right]\,\mathrm {d} \sigma _{1}\\&={\frac {1}{n!}}\int _{a}^{x}\left(x-t\right)^{n}f(t)\,\mathrm {d} t.\end{aligned}}}$$

• It has been shown that this statement holds true for the base case ${\displaystyle n=1}$.
• If the statement is true for ${\displaystyle n=k}$, then it has been shown that the statement holds true for ${\displaystyle n=k+1}$.
• 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 ${\displaystyle n\in \mathbb {Z} _{\geq 0}}$ is replaced by ${\displaystyle \alpha \in \mathbb {C} ,\ \Re (\alpha )>0}$, and the factorial is replaced by the gamma function. The two formulas agree when ${\displaystyle \alpha \in \mathbb {Z} _{\geq 0}}$.

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: 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

• Alan Beardon (2000). "Fractional calculus II". University of Cambridge.

• Maurice Mischler (2023). "About some repeated integrals and associated polynomials".