Infinite product

From Wikipedia, the free encyclopedia

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

$\prod_{n=1}^{\infty} a_n = a_1 \; a_2 \; a_3 \cdots$

is defined to be the limit of the partial products a₁a₂...a_n as n increases without bound. The product is said to converge when the limit exists and is not zero. Otherwise the product is said to diverge. The value zero is treated specially in order to obtain results analogous to those for infinite sums. If the product converges, then the limit of the sequence a_n as n increases without bound must be 1, while the converse is in general not true. Therefore, the logarithm log a_n will be defined for all but a finite number of n, and for those we have

$\log \prod_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} \log a_n$

with the product on the left converging if and only if the sum on the right converges. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products.

For products in which each $a_n\ge1$ , written as, for instance, $a n = 1 + p n$ , where $p_n\ge 0$ , the bounds

$1+\sum_{n=1}^{N} p_n \le \prod_{n=1}^{N} \left( 1 + p_n \right) \le \exp \left( \sum_{n=1}^{N}p_n \right)$

show that the infinite product converges precisely if the infinite sum of the p_n converges.

The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète and John Wallis (Wallis product):

$\frac{2}{\pi} = \frac{ \sqrt{2} }{ 2 } \cdot \frac{ \sqrt{2 + \sqrt{2}} }{ 2 } \cdot \frac{ \sqrt{2 + \sqrt{2 + \sqrt{2}}} }{ 2 } \cdots$

$\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots = \prod_{n=1}^{\infty} \left( \frac{ 4 \cdot n^2 }{ 4 \cdot n^2 - 1 } \right)$

[edit] Product representations of functions

Main article: Weierstrass factorization theorem

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. 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

$f(z) = z^m \; e^{\phi(z)} \; \prod_{n=1}^{\infty} \left(1 - \frac{z}{u_n} \right) \; \exp \left\lbrace \frac{z}{u_n} + \frac12\left(\frac{z}{u_n}\right)^2 + \cdots + \frac1{\lambda_n}\left(\frac{z}{u_n}\right)^{\lambda_n} \right\rbrace$

where λ_n are non-negative integers that can be chosen to make the product converge, and φ(z) is some uniquely determined analytic function (which means the term before the product will have no roots in the complex plane). The above factorization is not unique, since it depends on the choice of values for λ_n, and is not especially elegant. However, for most functions, there will be some minimum non-negative integer p such that λ_n = p gives a convergent product, called the canonical product representation. This p is called the rank of the canonical product. In the event that p = 0, this takes the form

$f(z) = z^m \; e^{\phi(z)} \; \prod_{n=1}^{\infty} \left(1 - \frac{z}{u_n}\right)$

This can be regarded as a generalization of the Fundamental Theorem of Algebra, since the product becomes finite and φ(z) is constant for polynomials.

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

Sine function	$\sin \pi z = \pi z \prod_{n=1}^{\infty} \left(1 - \frac{z^2}{n^2}\right)$	Euler - Wallis' formula for π is a special case of this.
Gamma function	$1 / \Gamma(z) = z \; \mbox{e}^{\gamma z} \; \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) \; \mbox{e}^{-z/n}$	Schlömilch
Weierstrass sigma function	$\sigma(z) = z\prod_{\omega \in \Lambda_{*}} \left(1-\frac{z}{\omega}\right)e^{\frac{1}{2\omega^2}z^2+\frac{1}{\omega}z}$	Here $Λ *$ is the lattice without the origin.
Riemann zeta function	$\zeta(z) = \prod_{n=1}^{\infty} \frac{1}{(1 - p_n^{-z})}$	Here p_n denotes the sequence of prime numbers.