Infinite product

An expression of the form

(*)

containing an infinite set of factors, numbers or functions, all of which are non-zero. An infinite product is said to be convergent if there exists a non-zero limit of the sequence of partial products

as . The value of the infinite product is the limit

and one writes

An infinite product converges if and only if the series

is convergent. Accordingly, the study of the convergence of infinite products is reduced to the study of the convergence of series. The infinite product (*) is said to be absolutely convergent if the infinite product

is convergent. A necessary and sufficient condition for absolute convergence of the infinite product (*) is absolute convergence of the series

An infinite product has the rearrangement property (i.e. its value is independent of the order of the factors) if and only if it is absolutely convergent.

The infinite product (*) with factors which are functions

defined, for example, in a domain of the complex -plane, converges uniformly in if the sequence of partial products converges uniformly in to a non-zero limit. A very important case in practical applications is when certain factors have zeros in such that at most a finite number of the zeros lie in any compact set . The concept of convergence is generalized as follows: The infinite product (*) is said to be (absolutely, uniformly) convergent inside if for any compact set there exists a number such that all the factors for , while the sequence of partial products

converges (absolutely, uniformly) on to a non-zero limit. If all factors are analytic functions in and if the infinite product converges uniformly inside , its limit is an analytic function in .

Infinite products were first encountered by F. Viète (1593) in his study of the quadrature of the circle. He represented the number analytically by the following infinite product:

Another representation of is due to J. Wallis (1665):

Infinite products with factors that are functions were encountered by L. Euler (1742); an example is

Infinite products are a principal tool in representing analytic functions with explicit indication of their zeros; for entire functions (cf. Entire function) they are the analogue of the factors of polynomials. See also Blaschke product; Weierstrass theorem on infinite products; Canonical product.

References

[1]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1 , MIR (1982) (Translated from Russian)
[2]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)
[3]	A.V. Bitsadze, "Fundamentals of the theory of analytic functions of a complex variable" , Moscow (1969) (In Russian)

Comments

See also Hadamard theorem on entire functions.

References

[a1]	J.B. Conway, "Functions of one complex variable" , Springer (1984)
[a2]	A.S.B. Holland, "Introduction to the theory of entire functions" , Acad. Press (1973)