A product integral is any product-based counterpart of the usual sum-based integral of calculus. The first product integral (Type I below) was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations.[1][2] Other examples of product integrals are the geometric integral (Type II below), the bigeometric integral (Type III below), and some other integrals of non-Newtonian calculus.[3][4][5]
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 image analysis[6] and in the study of growth/decay phenomena (e.g., in economic growth, bacterial growth, and radioactive decay).[7][8] The bigeometric integral, together with the bigeometric derivative, is useful in some applications of fractals,[9][10][11][12] and in the theory of elasticity in economics.[3][5][13]
This article adopts the "product" notation for product integration instead of the "integral" (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.
^
V. Volterra, B. Hostinský, Opérations Infinitésimales Linéaires, Gauthier-Villars, Paris (1938).
^
A. Slavík, Product integration, its history and applications, ISBN 80-7378-006-2, Matfyzpress, Prague, 2007.
^ abCite error: The named reference nnc was invoked but never defined (see the help page).
^
Michael Grossman. The First Nonlinear System of Differential And Integral Calculus, ISBN 0977117006, 1979.
^ abMichael Grossman. Bigeometric Calculus: A System with a Scale-Free Derivative, ISBN 0977117030, 1983.
^
Luc Florack and Hans van Assen."Multiplicative calculus in biomedical image analysis", Journal of Mathematical Imaging and Vision, doi:10.1007/s10851-011-0275-1, 2011.
^
Diana Andrada Filip and Cyrille Piatecki. "An overview on non-Newtonian calculus and its potential applications to economics", Applied Mathematics – A Journal of Chinese Universities, Volume 28, China Society for Industrial and Applied Mathematics, Springer, 2014.
^
Agamirza E. Bashirov, Emine Misirli, Yucel Tandogdu, and Ali Ozyapici."On modelling with multiplicative differential equations", Applied Mathematics – A Journal of Chinese Universities, Volume 26, Number 4, pages 425–428, doi:10.1007/s11766-011-2767-6, Springer, 2011.
^Marek Rybaczuk."Critical growth of fractal patterns in biological systems", Acta of Bioengineering and Biomechanics, Volume 1, Number 1, Wroclaw University of Technology, 1999.
^Marek Rybaczuk, Alicja Kedzia and Witold Zielinski (2001) "The concept of physical and fractal dimension II. The differential calculus in dimensional spaces", Chaos, Solitons, & FractalsVolume 12, Issue 13, October 2001, pages 2537–2552.
^
Dorota Aniszewska and Marek Rybaczuk (2005) "Analysis of the multiplicative Lorenz system", Chaos, Solitons & Fractals
Volume 25, Issue 1, July 2005, pages 79–90.
^
Fernando Córdova-Lepe. "The multiplicative derivative as a measure of elasticity in economics", TMAT Revista Latinoamericana de Ciencias e Ingeniería, Volume 2, Number 3, 2006.
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