# Product integral

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" $\prod$ notation for product integration instead of the "integral" $\int$ (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.

It is when applied to functions of a non-commutative 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]

## Basic definitions

The classical Riemann integral of a function $f:[a,b]\to\mathbb{R}$ can be defined by the relation

$\int_a^b f(x)\,dx = \lim_{\Delta x\to 0}\sum f(x_i)\,\Delta x,$

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

$\prod_a^b f(x)^{dx} = \lim_{\Delta x\to 0}\prod{f(x_i)^{\Delta x}} =\exp\left(\int_a^b \ln f(x) \, dx\right)$,

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 $\prod_{i=a}^b$ (with $i, a, b \in \mathbb{Z}$) and the multiplicative equivalent to the (normal/standard/additive) integral $\int_a^b dx$ (with $x \in [a,b]$):

 additive multiplicative discrete $\sum_{i=a}^b f(i)$ $\prod_{i=a}^b f(i)$ continuous $\int_a^b f(x) dx$ $\prod_a^b f(x) {}^{dx}$

It is very useful in stochastics where the log-likelihood (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:

$\ln \prod_a^b p(x)^{dx} = \int_a^b \ln p(x) \, dx$

### Type II

$\prod_a^b (1+f(x)\,dx) = \lim_{\Delta x\to 0} \prod (1+f(x_i)\,\Delta x)$

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 $f:[a,b]\to\mathbb{R}$:

$\prod_a^b (1+f(x)\,dx) =\exp\left(\int_a^b f(x) \, dx\right),$

which is not a multiplicative operator. However, this type of product integral is most useful when applied to matrix-valued 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
$\; \prod_a^b {f^*(x)^{dx}} = \prod_a^b \exp\left (\frac{f'(x)}{f(x)}\, dx\right ) = \frac{f(b)}{f(a)}$

where f * (x) is the "geometric derivative".

• Product rule
$\; (fg)^* = f^*g^*$
• Quotient rule
$\; (f/g)^* = f^*/g^*$
• Law of large numbers
$\; \sqrt[n] {X_1 X_2 \cdots X_n} \to \sideset{}{}\prod_x X^{\operatorname{pr}(x)\,dx} \text{ as }n \to \infty$
where X is a random variable with probability distribution pr(x)).
Compare with the standard Law of Large Numbers:
$\; \frac{X_1+X_2+\cdots+X_n}{n} \; \to \; \int X\, \operatorname{pr}(x)\,dx\text{ as }n \to \infty$

The above are for Type I Product integrals. Other types produce other results.

## References

1. ^ Multiplicative Calculus in Biomedical Image Analysis, Luc Florack, Hans van Assen, 2011, J Math Imaging Vis DOI 10.1007/s10851-011-0275-1
• 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, Gauthier-Villars, 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, Non-Newtonian Calculus, ISBN 0912938013, Lee Press, 1972.
• A. Slavík, Product integration, its history and applications, ISBN 80-7378-006-2, Matfyzpress, Prague, 2007.

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