Block, SkipCaptcha, Upload, Automoderated users, Check users, delete, edit, Moderators, move, oversight, protect, rollback
19,369
edits
Changes
Jump to navigation
Jump to search
mLine 5:
Line 5: − + Line 16:
Line 16: − + − + − + − + + + +
revealing ref
</math>
</math>
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 the infinite sum <math>\sum_{n=1}^{\infty} \ln a_n</math> converge.
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 representation of entire functions==
==Infinite Product representation of entire functions==
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'':
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]]
==References==
[[category:complex analysis]]
{{Reflist}}
[[Category:Mathematics]]
[[Category:Complex Analysis]]