Cauchy's formula for repeated integration

Cauchy's formula for repeated integration expresses an iterated integral as a single integral with a power-law kernel. For an integer order, the kernel is the distance to the upper limit raised to one less than that order, divided by the corresponding factorial.

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Fubini's theorem: A theorem that permits exchanging the order of integration under suitable conditions. Changing the order over a simplex collapses the nested integrals into the formula's power kernel.
Iterated integral: An integral formed by applying integration successively, often with one integral nested inside another. The formula's left side is an iterated integral with a fixed lower endpoint.
Fractional calculus: The study of derivatives and integrals whose orders need not be integers. Replacing the factorial with a gamma function extends the identity to fractional orders.
Augustin-Louis Cauchy: A French mathematician who helped establish rigorous foundations for analysis in the nineteenth century. The formula carries Cauchy's name and reflects his work on integral calculus.
Simplex: The generalization of a triangle or tetrahedron to any dimension, defined by convex combinations of its vertices. The ordered integration variables form a simplex whose cross-sections produce the kernel.
Antiderivative: A function whose derivative equals a specified function on an interval. Repeated integration constructs successive antiderivatives, with constants fixed by the lower endpoint.
Riemann–Liouville integral: A fractional integral defined by integrating against a power-law kernel weighted by a gamma factor. It generalizes the repeated-integration formula from integer to real or complex order.
Integral calculus: The branch of calculus concerned with accumulation, areas, and antiderivatives. Repeated integration is an operation within integral calculus.
Beta function: A special function defined by an integral and related to gamma functions by B(x,y)=Γ(x)Γ(y)/Γ(x+y). Its integral evaluates the power-weighted volume underlying repeated integration.
Definite integral: The signed accumulation of a function over an interval, defined through limits of sums. The formula represents the accumulated effect of repeated definite integration.