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).
For non-integer n it yields the definition of fractional integrals and (with n < 0) fractional derivatives.
Let f be a continuous function on the real line.
Then the nth repeated integral of f with base-point a,
is given by single integration
A proof is given by induction.
The base case with n = 1 is trivial, since it is equivalent to
}}\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
}}\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,
{\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 that the term within square bracket has n-times successive integration, and upper limit of outermost integral inside the square bracket is
Thus, comparing with the case for n = n and replacing
of the formula at induction step n = n with
Putting this expression inside the square bracket results in
}}\int _{a}^{x}(x-t)^{n}f(t)\,\mathrm {d} t.\end{aligned}}}
This completes the proof.
The Cauchy formula is generalized to non-integer parameters by the Riemann–Liouville integral, where
α ∈
( α ) > 0
, and the factorial is replaced by the gamma function.
The two formulas agree when
α ∈
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.