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Cauchy formula for repeated integration

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


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