Infinite product

An infinite product

$\prod _{n=1}^{\infty }a_{n}=a_{1}\;a_{2}\;a_{3}\cdots$

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 converges if the infinite sum $\sum _{n=1}^{\infty }\log a_{n}$ 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:

$f(z)=z^{m}\;e^{\phi (z)}\;\prod _{n=1}^{\infty }\left(1-{\frac {z}{\lambda _{n}}}\right)\;e^{\left[{\frac {z}{\lambda _{n}}}+{\frac {1}{2}}\left({\frac {z}{\lambda _{n}}}\right)^{2}+\cdots +{\frac {1}{m_{n}}}\left({\frac {z}{\lambda _{n}}}\right)^{m_{n}}\right]}$

where m is the multiplicity of the zero of f(z) at the origin, and φ(z) is some entire function.