Difference between revisions of "Infinite product"
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:<math> | :<math> | ||
f(z) = z^m \; e^{\phi(z)} \; \prod_{n=1}^{\infty} \left(1 - \frac{z}{\lambda_n} \right) \; | f(z) = z^m \; e^{\phi(z)} \; \prod_{n=1}^{\infty} \left(1 - \frac{z}{\lambda_n} \right) \; | ||
| − | + | e^{\left [ \frac{z}{\lambda_n} + \frac12\left(\frac{z}{\lambda_n}\right)^2 + \cdots + \frac1{m_n}\left(\frac{z}{\lambda_n}\right)^{m_n} \right ]} | |
</math> | </math> | ||
Revision as of 05:10, April 15, 2007
An infinite product
- <math>
\prod_{n=1}^{\infty} a_n = a_1 \; a_2 \; a_3 \cdots </math>
of a sequence of numbers a1, a2, a3, ... is defined to be the limit of the partial products a1a2...an as n goes to infinity. The infinite product is said to converge when the limit exists and is not zero. Otherwise the product is said to diverge.
Infinite Product representation of entire functions
Karl Weierstrass proved that every entire function f(z) with a divergent sequence (λn) of zeros, can be factored into an infinite product of the form:
- <math>
f(z) = z^m \; e^{\phi(z)} \; \prod_{n=1}^{\infty} \left(1 - \frac{z}{\lambda_n} \right) \; e^{\left [ \frac{z}{\lambda_n} + \frac12\left(\frac{z}{\lambda_n}\right)^2 + \cdots + \frac1{m_n}\left(\frac{z}{\lambda_n}\right)^{m_n} \right ]} </math>
where m is the multiplicity of the zero of f(z) at the origin, and φ(z) is some entire function.