Cauchy formula for repeated integration

The Cauchy formula for repeated integration allows one to compress $n$ antidifferentiations of a function into a single integral.

calar case

Let $f$ be a continuous function on the real line. Then the $n^\{th\}$ antidifferentiation of $f$,

:$f^\{\; [n]\; \}(x)\; =\; int\_\{0\}^xint\_0^\{sigma\_1\}cdotsint\_0^\{sigma\_\{n-1f(sigma\_\{n\})dsigma\_\{n\}cdots\; dsigma\_2\; dsigma\_1$,

is given by single integration

:$f^\{\; [n]\; \}(x)\; =\; frac\{1\}\{(n-1)!\}int\_0^xleft(x-y\; ight)^\{n-1\}f(y)dy$.

A proof is given by induction. Since $f$ is continuous, the base case is given by

:$frac\{d\}\{dx\}f^\{\; [1]\; \}(x)\; =\; frac\{d\}\{dx\}int\_0^xf(y)dy\; =\; f(x)$.

A little work shows that we also have

:$frac\{d\}\{dx\}f^\{\; [n]\; \}(x)\; =\; frac\{d\}\{dx\}frac\{1\}\{(n-1)!\}int\_0^xleft(x-y\; ight)^\{n-1\}f(y)dy\; =\; f^\{\; [n-1]\; \}(x)$.

Hence, $f^\{\; [n]\; \}(x)$ gives the $n^\{th\}$ antidifferentiation of $f(x)$.

References

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

ee also

*Fractional calculus