Infinite product
An expression of the form
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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
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as . The value of the infinite product is the limit
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and one writes
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An infinite product converges if and only if the series
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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
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is convergent. A necessary and sufficient condition for absolute convergence of the infinite product (*) is absolute convergence of the series
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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
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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
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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:
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Another representation of is due to J. Wallis (1665):
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Infinite products with factors that are functions were encountered by L. Euler (1742); an example is
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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) |
Infinite product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinite_product&oldid=13531