Infinite product

From Wikipedia for FEVERv2
Jump to navigation Jump to search

In mathematics, for a sequence of complex numbers a1, a2, a3, ... the infinite product Infinite product_sentence_0

is defined to be the limit of the partial products a1a2...an as n increases without bound. Infinite product_sentence_1

The product is said to converge when the limit exists and is not zero. Infinite product_sentence_2

Otherwise the product is said to diverge. Infinite product_sentence_3

A limit of zero is treated specially in order to obtain results analogous to those for infinite sums. Infinite product_sentence_4

Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. Infinite product_sentence_5

If the product converges, then the limit of the sequence an as n increases without bound must be 1, while the converse is in general not true. Infinite product_sentence_6

The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète (Viète's formula, the first published infinite product in mathematics) and John Wallis (Wallis product): Infinite product_sentence_7

Convergence criteria Infinite product_section_0

The product of positive real numbers Infinite product_sentence_8

converges to a nonzero real number if and only if the sum Infinite product_sentence_9

converges. Infinite product_sentence_10

This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products. Infinite product_sentence_11

The same criterion applies to products of arbitrary complex numbers (including negative reals) if the logarithm is understood as a fixed branch of logarithm which satisfies ln(1) = 0, with the proviso that the infinite product diverges when infinitely many an fall outside the domain of ln, whereas finitely many such an can be ignored in the sum. Infinite product_sentence_12

and by the limit comparison test it follows that the two series Infinite product_sentence_13

are equivalent meaning that either they both converge or they both diverge. Infinite product_sentence_14

Product representations of functions Infinite product_section_1

Main article: Weierstrass factorization theorem Infinite product_sentence_15

One important result concerning infinite products is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be factored into an infinite product of entire functions, each with at most a single root. Infinite product_sentence_16

In general, if f has a root of order m at the origin and has other complex roots at u1, u2, u3, ... (listed with multiplicities equal to their orders), then Infinite product_sentence_17

This can be regarded as a generalization of the fundamental theorem of algebra, since for polynomials, the product becomes finite and φ(z) is constant. Infinite product_sentence_18

In addition to these examples, the following representations are of special note: Infinite product_sentence_19

The last of these is not a product representation of the same sort discussed above, as ζ is not entire. Infinite product_sentence_20

Rather, the above product representation of ζ(z) converges precisely for Re(z) > 1, where it is an analytic function. Infinite product_sentence_21

By techniques of analytic continuation, this function can be extended uniquely to an analytic function (still denoted ζ(z)) on the whole complex plane except at the point z = 1, where it has a simple pole. Infinite product_sentence_22

See also Infinite product_section_2

Infinite product_unordered_list_0


Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Infinite product.