Cauchy formula for repeated integration

The Cauchy formula for repeated integration, named after Augustin Louis Cauchy, allows one to compress n antidifferentiations of a function into a single integral (cf. Cauchy's formula).

Scalar case

Let ƒ be a continuous function on the real line. Then the nth antiderivative of ƒ,

$f^{[n]}(x)=\int _{a}^{x}\int _{a}^{\sigma _{1}}\cdots \int _{a}^{\sigma _{n-1}}f(\sigma _{n})\,d\sigma _{n}\cdots \,d\sigma _{2}\,d\sigma _{1},$

is given by single integration

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

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

${\frac {d}{dx}}f^{[1]}(x)={\frac {d}{dx}}\int _{a}^{x}f(y)\,dy=f(x).$

A little work shows that we also have

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

Hence, ƒ^[n](x) gives the nth antiderivative of ƒ(x).

See also

• Fractional calculus

References

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

External links

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