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mLine 5:
Line 5: − + − + − + − \exp \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 ]+ + + + − + + + + +
revealing ref
</math>
</math>
of a [[sequence]] of numbers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... is defined to be the [[limit (mathematics)|limit]] of the partial products ''a''<sub>1</sub>''a''<sub>2</sub>...''a''<sub>''n''</sub> as ''n'' goes to infinity. The infinite product is said to [[convergence|converge]] when the limit exists and is not zero. Otherwise the product is said to [[diverge]].
of a [[sequence]] of terms ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... is defined to be the [[limit (mathematics)|limit]] of the partial products ''a''<sub>1</sub>''a''<sub>2</sub>...''a''<sub>''n''</sub> 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 representations of entire functions==
==Infinite Product representation of entire functions==
[[Karl Weierstrass]] proved that every [[entire function]] ''f''(''z'') with a divergent sequence (λ<sub>''n''</sub>) of zeros, can be factored into an infinite product of the form
[[Karl Weierstrass]] proved that every [[entire function]] ''f''(''z'') with a sequence (λ<sub>''n''</sub>) of zeros that does not [[limit point|accumulate]], can be factored into an infinite product of the form:
:<math>
:<math>
f(z) = z^m \; e^{\phi(z)} \; \prod_{n=1}^{\infty} \left(1 - \frac{z}{\lambda_n} \right) \;
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>
</math>
where ''m'' is the multiplicity of the zero of ''f''(''z'') at the origin, and φ(''z'') is some [[entire function]].
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]] <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>
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]]
==References==
{{Reflist}}
[[Category:Mathematics]]
[[Category:Complex Analysis]]