Product integral

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Product integrals are a multiplicative version of standard integrals of infinitesimal calculus. They were first developed by the mathematical biologist Vito Volterra 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.

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.

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[edit] 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 convergence and for the last type one also has to describe the partition of the domain for the limit)

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

[edit] Example

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

[edit] Results

Like standard calculus, product calculus has "multiplicative" analogs of standard results (for suitable f(x), a, b) like:

  • The fundamental theorem
\; \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
\; (fg)^* = f^*g^*
  • Quotient rule
\; (f/g)^* = f^*/g^*


  • Law of large Numbers
\; \sqrt[n] {X_1*X_2*...*X_n} \to \sideset{}{}\prod_{x}{X^{pr(x)dx}} \; 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+...+X_n}{n} \; \to \; \int X*pr(x)dx\; as \; n \to \infty

(note: the above are for Type I Product integrals. Other types produce other results).

[edit] References

  • 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
  • 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).

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