Infinite Product


Also found in: Dictionary, Wikipedia.

Infinite Product

 

the product of an infinite number of factors u1, u2,..., un,. . .—that is an expression of the form

An infinite product in which the factors are numbers is sometimes called an infinite numerical product. An infinite product cannot always be assigned a numerical value. If there exists a limit of the sequence of partial products

Pn = u1u2 . . . un

which is distinct from zero as n → ∞, then the infinite product is called convergent, and lim pn = p is its value. We write

Historically, the infinite product was first encountered in connection with problems concerning the calculation of the number π Thus, the 16th-century French mathematician F. Vieta obtained the formula

and the 17th-century English mathematician J. Wallis the formula

The infinite product acquired special importance after the work of L. Euler, who used the infinite product for the representation of functions. An example is the expansion of sin:

The expansion of functions into infinite products is analogous to the expansion of polynomials into linear factors; they are unusual in that they indicate all values of the independent variable for which the function vanishes.

For the convergence of an infinite product, it is necessary and sufficient that un = 0 for all numbers n, that uN< 0, starting with some number N, and that the series

converges. Thus, the study of the convergence of an infinite product is equivalent to the study of the convergence of this series.

REFERENCES

Fikhtengol’ts, G. M. Kurs differentsial’nogo i integral’nogo ischi-sleniia, vol. 2. Moscow-Leningrad, 1966.
Il’in, V. A., and E. G. Pozniak. Osnovy matematicheskogo analiza. Moscow, 1965.
References in periodicals archive ?
Yesilyurt [8] show the periodicity of signs of a large number of quotients of certain infinite products.
Yesilyurt, The periodicity of the signs of the coefficients of certain infinite products, Pacific J.
Szekers, The Taylor coefficients of certain infinite products, Acta Sci.
We begin with the well-known expansions of the theta functions as infinite products (Ref.
This section uses simple reflection arguments to derive infinite product formulas for the maps from circular domains to canonical circular and radial slit domains; see [34].
We will show that, for circle domains satisfying our separation criterion, the map can be represented by an infinite product formula.
The proof that f (z) defined by the (convergent) infinite product formula satisfies the boundary conditions in Lemma 3.
Thanks to Mviva's innovative content and marketing, their gateway is processing a larger volume of wireless transactions than the majority of network operator gateways, again proving the reliability and scalability of Infinite products.
Infinite products include the award-winning WAPLite WAP Server, the Infinite Enterprise WAP Server, and Infinite InterChange, the most powerful WAP e-mail solution available for corporate, ISP and mobile operator environments.
Infinite Products of Operators and Their Applications; proceedings
AVT has more than 75,000 systems installed worldwide, with 80 percent of Fortune 100 companies using the company's award winning CallXpress, RightFAX, MediaLinq or Infinite products and services.
This new agreement, coupled with our entire base of nationwide dealers, has presented Vodavi with an extraordinary opportunity to continue increasing market share for infinite products.