Difference between revisions of "Catalan's constant using Hurwitz zeta"
From specialfunctionswiki
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| − | + | ==Theorem== | |
| − | + | 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 [[Glaisher–Kinkelin constant]], and $\zeta'$ denotes the partial derivative of the [[Hurwitz zeta]] function with respect to the first argument. | 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== | |
| − | + | ||
| − | + | ==References== | |
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 08:01, 8 June 2016
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.
