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</math>

of a [[sequence]] of terms ''a''₁, ''a''₂, ''a''₃, ... is defined to be the [[limit (mathematics)|limit]] of the partial products ''a''₁''a''₂...''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.

of a [[sequence]] of terms ''a''₁, ''a''₂, ''a''₃, ... is defined to be the [[limit (mathematics)|limit]] of the partial products ''a''₁''a''₂...''a''_''n'' as ''n'' goes to infinity.  The infinite product converges if and only if the infinite sum <math>\sum_{n=1}^{\infty} \ln a_n</math> converge.

==Infinite Product representation of entire functions==

==Infinite Product representation of entire functions==

One spectacular result of the ''Weierstrass Factorization Theorem'' is the representation of the [[Riemann Zeta function]] <math>\zeta</math> as a product over its non-trivial zeros ''n'', known as the ''Hadamard Product'':

One spectacular result of the ''Weierstrass Factorization Theorem'' is the representation of the [[Riemann Zeta function]] <math>\zeta</math> as a product over its non-trivial zeros ''n'', known as the ''Hadamard Product'':

:<math>\zeta(z) = \frac{e^{\left [ ln 2 \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}}</math>

:<math>\zeta(z) = \frac{e^{\left [ ln 2 \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}}</math>

where <math>\gamma</math> is the [[Euler-Mascheroni constant]] and <math>\Gamma</math> is the [[Gamma function]].<ref>[http://mathworld.wolfram.com/HadamardProduct.html]</ref>

where <math>\gamma</math> is the [[Euler-Mascheroni constant]] and <math>\Gamma</math> is the [[Gamma function]].<ref>http://mathworld.wolfram.com/HadamardProduct.html</ref>

==References==

==References==

[[Category:Mathematics]]

[[Category:Mathematics]]

[[Category:Complex analysis]]

[[Category:Complex Analysis]]