For the case of , the product integral reduces exactly to the case of Lebesgue integration, that is, to classical calculus. Thus, the interesting cases arise for functions where is either some commutative algebra, such as a finite-dimensional matrix field, or if is a non-commutative algebra. The theories for these two cases, the commutative and non-commutative cases, have little in common. The non-commutative case is far more complicated; it requires proper path-ordering to make the integral well-defined.
The Volterra product integral is most useful when applied to matrix-valued functions or functions with values in a Banach algebra. When applied to scalars belonging to a non-commutative field, to matrixes, and to operators, i.e. to mathematical objects that don't commute, the Volterra integral splits in two definitions.[14]
The left product integral is
With this notation of left products (i.e. normal products applied from left)
The right product integral
With this notation of right products (i.e. applied from right)
Where is the identity matrix and D is a partition of the interval [a,b] in the Riemann sense, i.e. the limit is over the maximum interval in the partition. Note how in this case time ordering becomes evident in the definitions.
The product integral satisfies a collection of properties defining a one-parameter continuous group; these are stated in two articles showing applications: the Dyson series and the Peano–Baker series.
Commutative case
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The commutative case is vastly simpler, and, as a result, a large variety of distinct notations and definitions have appeared. Three distinct styles are popular in the literature. This subsection 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 adopted to impose some order in the field.
When the function to be integrated is valued in the real numbers, then the theory reduces exactly to the theory of Lebesgue integration.
Type I: Volterra integral
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The type I product integral corresponds to Volterra's original definition.[2][15][16] The following relationship exists for scalar functions:
Type II: Geometric integral
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which is called the geometric integral. The logarithm is well-defined if f takes values in the real or complex numbers, or if f takes values in a commutative field of commuting trace-class operators. This definition of the product integral is the continuous analog of the discreteproductoperator (with ) and the multiplicative analog to the (normal/standard/additive) integral (with ):
The type III product integral is called the bigeometric integral.
Basic results
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For the commutative case, the following results hold for the type II product integral (the geometric integral).
The geometric integral (type II above) plays a central role in the geometric calculus,[3][4][17] which is a multiplicative calculus. The inverse of the geometric integral, which is the geometric derivative, denoted , is defined using the following relationship:
When the integrand takes values in the real numbers, then the product intervals become easy to work with by using simple functions. Just as in the case of Lebesgue version of (classical) integrals, one can compute product integrals by approximating them with the product integrals of simple functions. The case of Type II geometric integrals reduces to exactly the case of classical Lebesgue integration.
The (type I) product integral was defined to be, roughly speaking, the limit of these products by Ludwig Schlesinger in a 1931 article.[which?]
Another approximation of the "Riemann definition" of the type I product integral is defined as
When is a constant function, the limit of the first type of approximation is equal to the second type of approximation.[19] Notice that in general, for a step function, the value of the second type of approximation doesn't depend on the partition, as long as the partition is a refinement of the partition defining the step function, whereas the value of the first type of approximation does depend on the fineness of the partition, even when it is a refinement of the partition defining the step function.
It turns out that[20] for any product-integrable function , the limit of the first type of approximation equals the limit of the second type of approximation. Since, for step functions, the value of the second type of approximation doesn't depend on the fineness of the partition for partitions "fine enough", it makes sense to define[21] the "Lebesgue (type I) product integral" of a step function as
where is the tagged partition corresponding to the step function . (In contrast, the corresponding quantity would not be unambiguously defined using the first type of approximation.)
This generalizes to arbitrary measure spaces readily. If is a measure space with measure, then for any product-integrable simple function (i.e. a conical combination of the indicator functions for some disjoint measurable sets ), its type I product integral is defined to be
since is the value of at any point of . In the special case where , is Lebesgue measure, and all of the measurable sets are intervals, one can verify that this is equal to the definition given above for that special case. Analogous to the theory of Lebesgue (classical) integrals, the Type I product integral of any product-integrable function can be written as the limit of an increasing sequence of Volterra product integrals of product-integrable simple functions.
Taking logarithms of both sides of the above definition, one gets that for any product-integrable simple function :
where we used the definition of integral for simple functions. Moreover, because continuous functions like can be interchanged with limits, and the product integral of any product-integrable function is equal to the limit of product integrals of simple functions, it follows that the relationship
holds generally for any product-integrable . This clearly generalizes the property mentioned above.
The Type I integral is multiplicative as a set function,[22] which can be shown using the above property. More specifically, given a product-integrable function one can define a set function by defining, for every measurable set ,
where denotes the indicator function of . Then for any two disjoint measurable sets one has
However, the Type I integral is notmultiplicative as a functional. Given two product-integrable functions , and a measurable set , it is generally the case that
Type II: Geometric integral
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If is a measure space with measure , then for any product-integrable simple function (i.e. a conical combination of the indicator functions for some disjoint measurable sets ), its type II product integral is defined to be
This can be seen to generalize the definition given above.
Taking logarithms of both sides, we see that for any product-integrable simple function :
where the definition of the Lebesgue integral for simple functions was used. This observation, analogous to the one already made for Type II integrals above, allows one to entirely reduce the "Lebesgue theory of type II geometric integrals" to the Lebesgue theory of (classical) integrals. In other words, because continuous functions like and can be interchanged with limits, and the product integral of any product-integrable function is equal to the limit of some increasing sequence of product integrals of simple functions, it follows that the relationship
holds generally for any product-integrable . This generalizes the property of geometric integrals mentioned above.
^
V. Volterra, B. Hostinský, Opérations Infinitésimales Linéaires, Gauthier-Villars, Paris (1938).
^ ab
A. Slavík, Product integration, its history and applications, ISBN 80-7378-006-2, Matfyzpress, Prague, 2007.
^ abc
M. Grossman, R. Katz, Non-Newtonian Calculus, ISBN 0-912938-01-3, Lee Press, 1972.
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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.
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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.
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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.
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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.
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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.
^Cherednikov, Igor Olegovich; Mertens, Tom; Van der Veken, Frederik (2 December 2019). Wilson Lines in Quantum Field Theory. Walter de Gruyter GmbH & Co KG. ISBN 9783110651690.
^Dollard, J. D.; Friedman, C. N. (1979). Product integration with applications to differential equations. Addison Wesley. ISBN 0-201-13509-4.
^Gantmacher, F. R. (1959). The Theory of Matrices. Vol. 1 and 2.
^A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı. Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 2008.
^A. Slavík, Product integration, its history and applications, p. 65. Matfyzpress, Prague, 2007. ISBN 80-7378-006-2.
^A. Slavík, Product integration, its history and applications, p. 71. Matfyzpress, Prague, 2007. ISBN 80-7378-006-2.
^A. Slavík, Product integration, its history and applications, p. 72. Matfyzpress, Prague, 2007. ISBN 80-7378-006-2.
^A. Slavík, Product integration, its history and applications, p. 80. Matfyzpress, Prague, 2007. ISBN 80-7378-006-2
^Gill, Richard D., Soren Johansen. "A Survey of Product Integration with a View Toward Application in Survival Analysis". The Annals of Statistics 18, no. 4 (December 1990): 1501—555, p. 1503.
External links
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Non-Newtonian calculus website
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 Dx-less Calculus
Notes On the Lax equation
Antonín Slavík, An introduction to product integration
Antonín Slavík, Henstock–Kurzweil and McShane product integration