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Material for impending article on Product Integrals (please don't edit!):
Product Integrals are a multiplicative version of standard integrals of infinitesimal calculus. They were first developed by the mathematical biologist Vito Volterra (May 3, 1860 - October 11, 1940) in the 1890s to solve simultaneous differential equations. Since then Product Integrals have found use in areas from epidemiology (the Kaplan-Meier estimator) to stochastic population dynamics (multigrals), analysis and even quantum mechanics.

Sadly, Product Integrals never really entered mainstream mathematics, probably due to the counterintuitive notation that Volterra used. To date, various versions of Product Calculus are regularly rediscovered and the bewildering range of terminology and notation continues to grow.

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. Also an arbitrary classification of types is adopted to impose some order in the field.

Basic Definitions
In their most basic form, standard integrals can be viewed as the limit of the series that calculates the area under the graph of the function f(x)

$$\int f(x)dx=\lim_{\Delta x\to 0}\sum{f(x_i)*{\Delta x}}$$

Product Integrals are similar except that instead of taking the limit of a sum, the limit of a product is taken instead.

$$\; \sideset{}{_a^b}\prod_{x\in\Re}{f(x)^{dx}}=\lim_{\Delta x\to 0}\prod{f(x_i)^{\Delta x}}$$

It can be thought of as a "continuous" rather than "discrete" product.

However, unlike standard calculus, there are several types of Product Integrals, which because of a lack of widely accepted terminology shall be arbitrarily designated Types I to III below.


 * Type I:

$$\; \sideset{}{_a^b}\prod_{x\in\Re}{f(x)^{dx}}=\lim_{\Delta x\to 0}\prod{f(x_i)^{\Delta x}} =exp(\int_a^b {ln(f(x))dx})$$


 * Type II:

$$\; \sideset{}{_a^b}\prod_{x\in\Re}{1+f(x)*{dx}}=\lim_{\Delta x\to 0}\prod{(1+f(x_i)*{\Delta x})}$$


 * Type III (dx-less):

$$\; \sideset{}{_a^b}\prod_{x\in\Re}{f(x)}=\lim_{\Delta x\to 0}\prod{f(x_i)} =exp(\int_a^b {ln(f(x))})$$

(note: conditions apply for what $$f(x), a, b$$ produce convergnece and for the last type you also have to describe the partition of the domain for the limit)

Arguments ($$x$$) in the above can be either real variables or matrices (see Gill refences below).

Example
$$\; \sideset{}{_1^2}\prod_{x\in\Re}{x^{dx}} =exp(\int_a^b {ln(x)dx})=exp(xln(x)-x)_1^2=4/e$$

Results
Like standard calculus, product calculus has "multiplicative" analogs of standard results like:

Product Derivative (Type I)

$$\;f^*(x)=exp\left (\frac{f'(x)}{f(x)}\right)$$

The Fundamental Theorem (Type I)

$$\; \sideset{}{_a^b}\prod_{x\in\Re}{f^*(x)^{dx}}=\sideset{}{_a^b}\prod_{x\in\Re}{exp\left (\frac{f'(x)}{f(x)}dx\right )}=\frac{f(b)}{f(a)}$$

where $$\;f^*(x)=exp(f'(x)/f(x))$$ is the product-derivative (or m-derivative).

Product Rule (Type I)

$$\; (fg)^* = f^*g^*$$

Quotient Rule (Type II)

$$\; (f/g)^* = f^*/g^*$$

Some References
Concerning Product Integrals and Exponentials W. P. Davis, J. A. Chatfield Proceedings of the American Mathematical Society, Vol. 25, No. 4 (Aug., 1970), pp. 743-747 doi:10.2307/2036741

Volterra, V., Hostinsky, B, Operations Infinitesimales Lineaires, Gauthier-Villars, Paris (1938).

Dollard, John D., Friedman, Charles, N., "Product Integrals and the Schrödinger Equation", Journ. Math. Phys. 18 #8,1598-1607 (1977).

R L Hudson and S Pulmannova Double Product Integrals and Enriquez Quantisation of Lie Algebras II Letters in Math. Physics (2005) 72 211-224

http://www.math.nyu.edu/~neylon/files/inequality2.pdf

http://graham.main.nc.us/~bhammel/SPDER/lax.html

http://www.math.uu.nl/people/gill/Preprints/prod_int_1.pdf

http://wiki.contextgarden.net/Product_integral

Product Integral in Other Laguages
Russian: Интегральное произведение

Ukrainian: інтегральний добуток

Finnish: tulointegraali

Danish: produckintegral

Catalan: integral producte