Infinite product

In mathematics, for a sequence of numbers ''a''₁, ''a''₂, ''a''₃, ... the infinite product :

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

is defined to be the limit (mathematics) of the partial products ''a''₁''a''₂...''a''_''n'' as ''n'' goes to infinity. 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 get results analogous to those for Infinite series. If the product converges, then the limit of the sequence ''a''_''n'' as ''n'' goes to infinity must be 1 while the converse is in general not true. Therefore, the logarithm log ''a''_''n'' will be defined for all but finitely many ''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 does. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products. The best known examples of infinite products are probably some of the formulae for pi, such as the following two products, respectively by Vi�te and Wallis:

\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

==Product representations of functions== One important result concerning infinite products is that every entire function ''f''(''z'') (i.e., every function that is holomorphic function over the entire complex number) can be factored into an infinite product of entire functions each with at most a single zero. In general, if ''f'' has a zero of order ''m'' at the origin and has other complex zeros at ''u''₁, ''u''₂, ''u''₃, ... (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} + \left(\frac{z}{u_n}\right)^2 + \cdots + \left(\frac{z}{u_n}\right)^{\lambda n} \right\rbrace

where λ''n'' are positive integers that can be chosen to make the series converge, and φ(''z'') is some uniquely determined analytic function (which means the term before the product will have no zeros in the complex plane). The above factorization is not unique, since it depends on the choice of λ''n''s, and is not especially elegant. For most functions, though, there will be some minimum positive integer ''p'' such that λ''n'' = ''p'' gives a convergent product, called the canonical product representation, and in the even that ''p'' = 1, 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 for polynomials the product becomes finite and φ(''z'') is constant. Aside from these, 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 product is a special case of this.
Gamma function	$1 / \Gamma(z) = z \; \mbox{e}^{\gamma z} \; \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) \; \mbox{e}^{-z/n}$	Schl�milch
Riemann zeta function	$\zeta(z) = \prod_{n=1}^{\infty} \frac{1}{(1 - p_n^{-z})}$	Euler - Here ''p''_''n'' denotes the sequence of prime number.

Note the last of these is not a product representation of the same sort discussed above, as ζ is not entire. Mathematical analysis

Infinite product

This article is probably very non-ideal, but when I needed this material a while ago it was hard to find, and so I figured wikipedia would be a good place to hold it. Please feel free to change it around and make it more encyclopedia-like, even more so than usual. :) ---- We probably need a little caveat about negative numbers: the log formula won't work without problems in that case. Maybe use a different log which is defined on the negative reals? --AxelBoldt

See other meanings of words starting from letter:

I

IA | IB | IC | ID | IE | IF | IG | IH | IJ | IK | IL | IM | IN | IO | IP | IR | IS | IT | IU | IW | IX | IY | IZ |

Words begining with Infinite_product:

Infinite_product
Infinite_product