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
$$\xi(z)=\dfrac{z}{2}(z-1)\pi^{-\frac{z}{2}}\Gamma\left(\dfrac{z}{2}\right)\zeta(s),$$
+
$$\xi(z)=\dfrac{z}{2}(z-1)\pi^{-\frac{z}{2}}\Gamma\left(\dfrac{z}{2}\right)\zeta(z),$$
where $\Gamma$ denotes the [[gamma function]] and $\zeta$ denotes the [[Riemann zeta function]].
+
where $\pi$ denotes [[pi]], $\Gamma$ denotes [[gamma]], and $\zeta$ denotes [[Riemann zeta]].
  
[[File:Complex Riemann Xi.jpg|500px]]
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<div align="center">
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<gallery>
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File:Complex Riemann Xi.jpg|Domain coloring of $\xi$.
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</gallery>
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</div>
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=Properties=
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[[Functional equation for Riemann xi]]<br />
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=References=
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* {{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 $(7)$
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[[Category:SpecialFunction]]

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.

Properties

Functional equation for Riemann xi

References