Infinite product

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...a_n 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 divergent sequence (λ_n) 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.