Difference between revisions of "Infinite product"
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==Infinite Product representation of entire functions== | ==Infinite Product representation of entire functions== | ||
| − | [[Karl Weierstrass]] proved that every [[entire function]] ''f''(''z'') with a sequence (λ<sub>''n''</sub>) of zeros that does | + | [[Karl Weierstrass]] proved that every [[entire function]] ''f''(''z'') with a sequence (λ<sub>''n''</sub>) of zeros that does not have a [[limit point]], can be factored into an infinite product of the form: |
:<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) \; | ||
Revision as of 23:15, April 16, 2007
An infinite product
- <math>
\prod_{n=1}^{\infty} a_n = a_1 \; a_2 \; a_3 \cdots </math>
of a sequence of terms a1, a2, a3, ... is defined to be the limit of the partial products a1a2...an as n goes to infinity. The infinite product converges if and only if the the infinite sum <math>\sum_{n=1}^{\infty} \ln a_n</math> converge.
Infinite Product representation of entire functions
Karl Weierstrass proved that every entire function f(z) with a sequence (λn) of zeros that does not have a limit point, 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.