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 f be a continuous function on the real line. Then the nth repeated integral of f with basepoint a, is given by single integration Proof A proof is given by induction. The base case with n=1 is trivial, since it is equivalent to: Now, suppose this is true for n, and let us prove it for n+1. Firstly, using the Leibniz integral rule, note that Then, applying the induction hypothesis, This completes the proof. Generalizations and applications The Cauchy formula is generalized to non-integer parameters by the Riemann-Liouville integral, where is replaced by , and the factorial is replaced by the gamma function. The two formulas agree when . Both the Cauchy formula and the Riemann-Liouville integral are generalized to arbitrary dimension 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. References 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). External links Augustin-Louis Cauchy Integral calculus Theorems in analysis