Difference between revisions of "Infinite product"
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| − | of a [[sequence]] of | + | 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 | + | ==Infinite Product representation of entire functions== |
| − | [[Karl Weierstrass]] proved that every [[entire function]] ''f''(''z'') with a | + | [[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> | ||
| − | [[ | + | ==References== |
| + | {{Reflist}} | ||
| + | |||
| + | [[Category:Mathematics]] | ||
| + | [[Category:Complex Analysis]] | ||
Latest revision as of 17:04, September 6, 2019
An infinite product
- <math>
\prod_{n=1}^{\infty} a_n = a_1 \; a_2 \; a_3 \cdots </math>
of a sequence of terms a1, a2, a3, ... is defined to be the limit of the partial products a1a2...an 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
Karl Weierstrass proved that every entire function f(z) with a sequence (λn) of zeros that does not accumulate, can be factored into an infinite product of the form:
- <math>
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>
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.[1]