Difference between revisions of "Hurwitz zeta"

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The Hurwitz zeta function is a generalization of the [[Riemann zeta function]] defined initially for $\mathrm{Re}(s)>1$ and $\mathrm{Re}(a)>0$ by  
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$$\zeta(s,a)= \displaystyle\sum_{n=0}^{\infty} \dfrac{1}{(n+a)^s}.$$
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The Hurwitz zeta function is a generalization of the [[Riemann zeta]] function defined initially for $\mathrm{Re}(s)>1$ and $\mathrm{Re}(a)>0$ by  
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$$\zeta(z,q)= \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{(k+q)^z}.$$
  
 
=Properties=
 
=Properties=
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[[Hurwitz zeta absolute convergence]]<br />
<strong>Theorem:</strong> The function $\zeta(s,a)$ is [[absolutely convergent]] for all $s$ with $\mathrm{Re}(s)>1$ and $a$ with $\mathrm{Re}(a)>0$..
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[[Relationship between Hurwitz zeta and gamma function]]<br />
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[[Relation between polygamma and Hurwitz zeta]]<br />
<strong>Proof:</strong>
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[[Bernoulli polynomial and Hurwitz zeta]]<br />
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[[Catalan's constant using Hurwitz zeta]]<br />
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=See Also=
<strong>Theorem:</strong> The function $\zeta(s,a)$ is [[analytic]] for all $s$ except for a simple pole at $s=1$ with [[residue]] $1$.
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[[Riemann zeta]]<br />
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<strong>Proof:</strong> █
 
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{{:Relationship between Hurwitz zeta and gamma function}}
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=References=
 
 
{{:Bernoulli polynomial and Hurwitz zeta}}
 
{{:Catalan's constant using Hurwitz zeta}}
 
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 01:27, 21 December 2016

The Hurwitz zeta function is a generalization of the Riemann zeta function defined initially for $\mathrm{Re}(s)>1$ and $\mathrm{Re}(a)>0$ by $$\zeta(z,q)= \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{(k+q)^z}.$$

Properties

Hurwitz zeta absolute convergence
Relationship between Hurwitz zeta and gamma function
Relation between polygamma and Hurwitz zeta
Bernoulli polynomial and Hurwitz zeta
Catalan's constant using Hurwitz zeta

See Also

Riemann zeta

References