Infinite product

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...a_n 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 (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.