Expressing certain multiple integrals as single integrals

Abstract

Repeated indefinite integration (variable upper limit) of a given function arises in many contexts both within mathematics proper and in applications of mathematics. If the lower limits are all the same, such integrals can be expressed as a single integral. This fact is, for example, the starting point for the development of the so‐called fractional calculus (differentiation and integration of non‐integer order). If the lower limits are not all equal, such integrals can be expressed as a single integral together with a polynomial ‘correction term’ added to the single integral. The proof of the main result also provides an interesting and natural application of the fundamental existence and uniqueness theorem for initial value problems for ordinary differential equations.