Difference between revisions of "Riemann xi"
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The Riemann $\xi$ function is defined by the formula | The Riemann $\xi$ function is defined by the formula | ||
$$\xi(z)=\dfrac{z}{2}(z-1)\pi^{-\frac{z}{2}}\Gamma\left(\dfrac{z}{2}\right)\zeta(z),$$ | $$\xi(z)=\dfrac{z}{2}(z-1)\pi^{-\frac{z}{2}}\Gamma\left(\dfrac{z}{2}\right)\zeta(z),$$ | ||
| − | where $\Gamma$ denotes | + | where $\pi$ denotes [[pi]], $\Gamma$ denotes [[gamma]], and $\zeta$ denotes [[Riemann zeta]]. |
| − | + | <div align="center"> | |
| + | <gallery> | ||
| + | File:Complex Riemann Xi.jpg|Domain coloring of $\xi$. | ||
| + | </gallery> | ||
| + | </div> | ||
=Properties= | =Properties= | ||
| − | < | + | [[Functional equation for Riemann xi]]<br /> |
| − | + | ||
| − | + | =References= | |
| − | + | * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Functional equation for Riemann zeta with cosine|next=Functional equation for Riemann xi}}: § Introduction $(7)$ | |
| − | + | ||
| − | + | [[Category:SpecialFunction]] | |
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Latest revision as of 15:31, 18 March 2017
The Riemann $\xi$ function is defined by the formula $$\xi(z)=\dfrac{z}{2}(z-1)\pi^{-\frac{z}{2}}\Gamma\left(\dfrac{z}{2}\right)\zeta(z),$$ where $\pi$ denotes pi, $\Gamma$ denotes gamma, and $\zeta$ denotes Riemann zeta.
Domain coloring of $\xi$.
Properties
Functional equation for Riemann xi
References
- 1930: Edward Charles Titchmarsh: The Zeta-Function of Riemann ... (previous) ... (next): § Introduction $(7)$
