Product integral

Product integrals are a multiplicative 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.

Product integrals have not entered mainstream mathematics, probably due to the counterintuitive notation that Volterra used. To date, various versions of Product Calculus are regularly rediscovered and the 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. An arbitrary classification of types is also adopted to impose some order in the field.

Basic definitions

The classical Riemann integral of a function $f:[a,b]\to \mathbb {R}$ is 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.

There are several types of product integrals. The most popular ones are the following:

Type I

$\prod _{a}^{b}f(x)^{dx}=\lim _{\Delta x\to 0}\prod {f(x_{i})^{\Delta x}}=\exp \left(\int _{a}^{b}\ln f(x)\,dx\right)$

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

$\prod _{a}^{b}(1+f(x)\,dx)=\exp \left(\int _{a}^{b}f(x)\,dx\right)$

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

Like standard calculus, product calculus has "multiplicative" analogs of standard results (for suitable ƒ(x), a, b). (See Multiplicative calculus#Multiplicative derivatives.)

• 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 product derivative (or multiplicative 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.

See also

• Multiplicative calculus
• List of derivatives and integrals in alternative calculi
• Ordered exponential

References

• 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.

External links

• 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