Infinite product
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In mathematics, for a sequence of numbers a1, a2, a3, ... the infinite product
- <math>
\prod_{n=1}^{\infty} a_n = a_1 \; a_2 \; a_3 \cdots <math>
is defined to be the limit of the partial products a1a2...an 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 sums. If the product converges, then the limit of the sequence an as n goes to infinity must be 1 while the converse is in general not true. Therefore, the logarithm log an will be defined for all but finitely many n, and for those we have
- <math>\log \prod_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} \log a_n<math>
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 π, such as the following two products, respectively by Viète and Wallis:
<math>\frac{2}{\pi} = \frac{ \sqrt{2} }{ 2 } \cdot \frac{ \sqrt{2 + \sqrt{2}} }{ 2 } \cdot \frac{ \sqrt{2 + \sqrt{2 + \sqrt{2}}} }{ 2 } \cdots<math>
<math>\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<math>
Product representations of functions
One important result concerning infinite products is that every entire function f(z) (i.e., 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 zero. In general, if f has a zero of order m at the origin and has other complex zeros at u1, u2, u3, ... (listed with multiplicities equal to their orders) then
- <math>
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 <math>
where λn 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 λns, and is not especially elegant. For most functions, though, there will be some minimum positive integer p such that λn = p gives a convergent product, called the canonical product representation, and in the even that p = 1, this takes the form
- <math>
f(z) = z^m \; e^{\phi(z)} \; \prod_{n=1}^{\infty} \left(1 - \frac{z}{u_n}\right) <math>
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:
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Sine function |
<math>\sin \pi z = \pi z \prod_{n=1}^{\infty} \left(1 - \frac{z^2}{n^2}\right)<math> |
Euler - Wallis' formula for π is a special case of this. |
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<math>1 / \Gamma(z) = z \; \mbox{e}^{\gamma z} \; \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) \; \mbox{e}^{-z/n}<math> |
Schlömilch | |
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<math>\zeta(z) = \prod_{n=1}^{\infty} \frac{1}{(1 - p_n^{-z})}<math> | Euler - Here pn denotes the sequence of prime numbers. |
Note the last of these is not a product representation of the same sort discussed above, as ζ is not entire.ja:総乗