Difference between revisions of "Catalan's constant using Hurwitz zeta"
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<strong>[[Catalan's constant using Hurwitz zeta|Proposition]]:</strong> The following formula holds: | <strong>[[Catalan's constant using Hurwitz zeta|Proposition]]:</strong> The following formula holds: | ||
$$K=\dfrac{\pi}{24} -\dfrac{\pi}{2}\log(A)+4\pi \zeta' \left(-1 , \dfrac{1}{4} \right),$$ | $$K=\dfrac{\pi}{24} -\dfrac{\pi}{2}\log(A)+4\pi \zeta' \left(-1 , \dfrac{1}{4} \right),$$ | ||
| − | where $K$ is [[Catalan's constant]], $A$ is the [[ | + | 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. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
</div> | </div> | ||
</div> | </div> | ||
Revision as of 01:18, 21 March 2015
Proposition: 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.
Proof: █
