Product integral

The expression "product integral" is used informally for referring to any product-based counterpart of the usual sum-based integral of classical calculus. The first product integral was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations. (Please see "Type II" below.) Other examples of product integrals are the geometric integral ("Type I" below), the bigeometric integral, and some other integrals of non-Newtonian calculus.

Product integrals have found use in areas from epidemiology (the Kaplan–Meier estimator) to stochastic population dynamics using multiplication integrals (multigrals), analysis and quantum mechanics. The geometric integral, together with the geometric derivative, is useful in biomedical image analysis.

This article adopts the "product" ${\displaystyle \prod }$ notation for product integration instead of the "integral" ${\displaystyle \int }$ (usually modified by a superimposed "times" symbol or letter P) favoured by Volterra and others. An arbitrary classification of types is also adopted to impose some order in the field.

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