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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>

[[category:mathematics]]

[[category:complex analysis]]

 

[[Category:Mathematics]]

[[Category:Complex analysis]]