Difference between revisions of "Hurwitz zeta"
From specialfunctionswiki
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| − | The Hurwitz zeta function is defined for $\mathrm{Re}(s)>1$ | + | The Hurwitz zeta function is defined for $\mathrm{Re}(s)>1$ and $\mathrm{Re}(a)>0$ by |
$$\zeta(s,a)= \displaystyle\sum_{n=0}^{\infty} \dfrac{1}{(n+a)^s}.$$ | $$\zeta(s,a)= \displaystyle\sum_{n=0}^{\infty} \dfrac{1}{(n+a)^s}.$$ | ||
=Properties= | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The function $\zeta(s,a)$ is [[absolutely convergent]] for all $s$ with $\mathrm{Re}(s)>1$ and $a$ with $\mathrm{Re}(a)>0$.. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
{{:Bernoulli polynomial and Hurwitz zeta}} | {{:Bernoulli polynomial and Hurwitz zeta}} | ||
{{:Catalan's constant using Hurwitz zeta}} | {{:Catalan's constant using Hurwitz zeta}} | ||
Revision as of 04:29, 12 April 2015
The Hurwitz zeta function is defined for $\mathrm{Re}(s)>1$ and $\mathrm{Re}(a)>0$ by $$\zeta(s,a)= \displaystyle\sum_{n=0}^{\infty} \dfrac{1}{(n+a)^s}.$$
Properties
Theorem: The function $\zeta(s,a)$ is absolutely convergent for all $s$ with $\mathrm{Re}(s)>1$ and $a$ with $\mathrm{Re}(a)>0$..
Proof: █
Theorem
The following formula holds: $$B_n(x)=-n \zeta(1-n,x),$$ where $B_n$ denotes the Bernoulli polynomial and $\zeta$ denotes the Hurwitz zeta function.
Proof
References
Theorem
The following formula holds: $$K=\dfrac{\pi}{24} -\dfrac{\pi}{2}\log(A)+4\pi \zeta' \left(-1 , \dfrac{1}{4} \right),$$ where $K$ is Catalan's constant, $A$ is the Glaisher–Kinkelin constant, and $\zeta'$ denotes the partial derivative of the Hurwitz zeta function with respect to the first argument.
