Product integral

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The expression "product integral" is used informally for referring to any product-based counterpart of the usual sum-based integral of classical calculus. The first product integral was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations.[1][2] (Please see "Type II" below.) Other examples of product integrals are the geometric integral ("Type I" below), the bigeometric integral, and some other integrals of non-Newtonian calculus.[3]

Product integrals have found use in areas from epidemiology (the Kaplan–Meier estimator) to stochastic population dynamics using multiplication integrals (multigrals), analysis and quantum mechanics. The geometric integral, together with the geometric derivative, is useful in biomedical image analysis.[4]

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.

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.

Roughly speaking, 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 analog of the discrete product operator \prod_{i=a}^b (with i, a, b \in \mathbb{Z}) and the multiplicative analog 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 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 Type II product integral corresponds to Volterra's original definition.[2][5][6] 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. (So the concepts of product integral and multiplicative integral are not the same). The Volterra 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).

For scalar functions, the derivative in the Volterra system is the logarithmic derivative, and so the Volterra system is not a multiplicative calculus and is not a non-Newtonian calculus.[2]


  • Basic results
\; \prod_a^b {c^{dx}} = c^{b-a}
\; \prod_a^b {(f(x)^k)^{dx}} = (\prod_a^b{f(x)^{dx}})^k
\; \prod_a^b {(c^{f(x)})^{dx}} = c^{\int_a^b \! f(x)dx}

The geometric integral (Type I above) plays a central role in the geometric calculus,[3] 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^{dF(x)} \text{ as }n \to \infty
where X is a random variable with probability distribution F(x).
Compare with the standard Law of Large Numbers:
\; \frac{X_1+X_2+\cdots+X_n}{n} \; \to \; \int X\, dF(x)\text{ as }n \to \infty

The above are for the Type I product integral. Other types produce other results.

See also


  1. V. Volterra, B. Hostinský, Opérations Infinitésimales Linéaires, Gauthier-Villars, Paris (1938).
  2. 2.0 2.1 2.2 A. Slavík, Product integration, its history and applications, ISBN 80-7378-006-2, Matfyzpress, Prague, 2007.
  3. 3.0 3.1 M. Grossman, R. Katz, Non-Newtonian Calculus, ISBN 0-912938-01-3, Lee Press, 1972.
  4. Luc Florack, Hans van Assen. "Multiplicative Calculus in Biomedical Image Analysis", Journal of Math Imaging and Vision, doi:10.1007/s10851-011-0275-1, 2011.
  5. J. D. Dollard, C. N. Friedman, Product integration with applications to differential equations, Addison Wesley Publishing Company, 1979.
  6. F.R. Gantmacher (1959) The Theory of Matrices, volumes 1 and 2.
  • 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.
  • 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.

External links