 Product integral

Product integrals are a counterpart of standard integrals of infinitesimal calculus. They were first developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations. Since then, product integrals have found use in areas from epidemiology (the Kaplan–Meier estimator) to stochastic population dynamics (multigrals), analysis and quantum mechanics.
Examples of product integrals are the geometric integral (see below) and the bigeometric integral. (Although those two integrals are multiplicative, the concepts of product integral and multiplicative integral are not the same).
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.
It is when applied to functions of a noncommutative variable such as matrix functions that product integrals offers something different  In the commutative case, calculations can be done using workarounds with the usual additive calculus operators.^{[1]}
Contents
Basic definitions
The classical Riemann integral of a function can be defined by the relation
where the limit is taken over all partitions of interval [a,b] whose norm approach zero.
Product integrals are similar, but take the limit of a product instead of the limit of a sum. They can be thought of as "continuous" versions of "discrete" products.
The most popular product integrals are the following:
Type I
 ,
which is called the "geometric integral" and is a multiplicative operator.
This definition of the product integral is the continuous equivalent of the discrete product operator (with ) and the multiplicative equivalent to the (normal/standard/additive) integral (with ):
additive multiplicative discrete continuous It is very useful in stochastics where the loglikelihood (i.e. the logarithm of a product integral of independent random variables) equals the integral of the log of the these (infinitesimally many) random variables:
Type II
Under these definitions, a real function is product integrable if and only if it is Riemann integrable. There are other more general definitions such as the Lebesgue product integral, Riemann–Stieltjes product integral, or Henstock–Kurzweil product integral.
The second type corresponds to Volterra's original definition. The following relationship exists for scalar functions :
which is not a multiplicative operator. However, this type of product integral is most useful when applied to matrixvalued functions or functions with values in a Banach algebra, where the last equality is no longer true (see the references below).
Results
The geometric integral (Type I above) plays a central role in the "geometric calculus", which is a multiplicative calculus.
 The fundamental theorem
where f ^{*} (x) is the "geometric derivative".
 Product rule
 Quotient rule
 Law of large numbers
 where X is a random variable with probability distribution pr(x)).
 Compare with the standard Law of Large Numbers:
The above are for Type I Product integrals. Other types produce other results.
See also
References
 ^ Multiplicative Calculus in Biomedical Image Analysis, Luc Florack, Hans van Assen, 2011, J Math Imaging Vis DOI 10.1007/s1085101102751
 A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı. Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 2008.
 W. P. Davis, J. A. Chatfield, Concerning Product Integrals and Exponentials, Proceedings of the American Mathematical Society, Vol. 25, No. 4 (Aug., 1970), pp. 743–747, doi:10.2307/2036741.
 V. Volterra, B. Hostinský, Opérations Infinitésimales Linéaires, GauthierVillars, Paris (1938).
 J. D. Dollard, C. N. Friedman, Product integrals and the Schrödinger Equation, Journ. Math. Phys. 18 #8,1598–1607 (1977).
 J. D. Dollard, C. N. Friedman, Product integration with applications to differential equations, Addison Wesley Publishing Company, 1979.
 M. Grossman, R. Katz, NonNewtonian Calculus, ISBN 0912938013, Lee Press, 1972.
 A. Slavík, Product integration, its history and applications, ISBN 8073780062, Matfyzpress, Prague, 2007.
External links
 Richard Gill, Product Integration
 Richard Gill, Product Integral Symbol
 David Manura, Product Calculus
 Tyler Neylon, Easy bounds for n!
 An Introduction to Multigral (Product) and Dxless Calculus
 Notes On the Lax equation
 Antonín Slavík, An introduction to product integration
 Antonín Slavík, Henstock–Kurzweil and McShane product integration
Categories: Integrals
 Multiplication
 NonNewtonian calculus
Wikimedia Foundation. 2010.