Infinite product

An infinite product

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

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 infinite sum $\sum _{n=1}^{\infty }\ln a_{n}$ converge.

Infinite Product representation of entire functions

Karl Weierstrass proved that every entire function f(z) with a sequence (λ_n) of zeros that does not accumulate, 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.

One spectacular result of the Weierstrass Factorization Theorem is the representation of the Riemann Zeta function $\zeta$ as a product over its non-trivial zeros n, known as the Hadamard Product:

$\zeta (z)={\frac {e^{\left[ln2\pi -1-{\frac {\gamma }{2}}\right]z}}{2(z-1)\Gamma (1+{\frac {z}{2}})}}\prod _{n}(1-{\frac {z}{n}})e^{\frac {z}{n}}$

where $\gamma$ is the Euler-Mascheroni constant and $\Gamma$ is the Gamma function.