Fonksyon zeta Riemann

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Ale nan: Navigasyon, Fouye

Se yon fonksyon matematik rele pa matematisyen alman Bernhard Riemann.

Fonksyon ζ(s) nan plan konplèks

Kontni

[edite] Definsiyon

Li defini pa seri sa :

\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \!

Fonksyon zeta Riemann zeta pou reyèl s > 1


[edite] Euler

Lyen ki egziste ant fonksyon zeta ak nonm premye yo te dekouvè pa matematisyen Leonhard Euler :

\begin{align} \zeta(s)=\sum_{n\geq 1}\frac{1}{n^s}& = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} \\ & = \left(1 + \frac{1}{2^s} + \frac{1}{4^s} + \cdots \right) \left(1 + \frac{1}{3^s} + \frac{1}{9^s} + \cdots \right) \cdots \left(1 + \frac{1}{p^s} + \frac{1}{p^{2s}} + \cdots \right) \cdots, \end{align} \!

p ap deziyen yon nonm premye.Seri sa genyen yon sans lè pati reyèl Re(s) > 1.

[edite] Propryete


[edite] Kèk valè pou konnen

\zeta(0) = -1/2,\!
\zeta(1/2) \approx -1.4603545088095868,\!
\zeta(1) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty;\! Seri amonik.
\zeta(3/2) \approx 2.612;\! itilize nan kalkil tanperati kritik pou kondansa Bose–Einstein nan domèn fizik
\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} \approx 1.645;\! demonstrasyon egalite sa rele tou pwoblèm Basel.
\zeta(5/2) \approx 1.341.\!
\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots \approx 1.202;\! relekonstant Apéry.
\zeta(7/2) \approx 1.127\!
\zeta(4) = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90} \approx 1.0823;\! lwa Stefan–Boltzmann epi apwoksimasyon Wien nan domèn fizik.

[edite] Fonksyon - analiz

Fonksyon zeta an satisfèt ekwasyon fonksyonèl sa tou :

\zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s) \!


[edite] Lyen

Rekipere depi « http://ht.wikipedia.org/wiki/Fonksyon_zeta_Riemann »
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