Difference between revisions of "Catalan's constant"
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=Properties= | =Properties= | ||
{{:Catalan's constant using Dirichlet beta}} | {{:Catalan's constant using Dirichlet beta}} | ||
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| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>[[Catalan's constant using Legendre chi]]:</strong> The following formula holds: | ||
| + | $$K=-i\chi_2(i),$$ | ||
| + | where $K$ is [[Catalan's constant]] and $\chi$ denotes the [[Legendre chi]] function. | ||
| + | where | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | {{:Catalan's constant using Hurwitz zeta}} | ||
Revision as of 01:17, 21 March 2015
Catalan's constant is $$G=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)^2} = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 \ldots.$$ This means that Catalan's constant can be expressed as $\beta(2)$ where $\beta$ is the Dirichlet beta function.
Properties
Theorem
The following formula holds: $$K=\beta(2),$$ where $K$ is Catalan's constant and $\beta$ denotes the Dirichlet beta function.
Proof
References
Catalan's constant using Legendre chi: The following formula holds: $$K=-i\chi_2(i),$$ where $K$ is Catalan's constant and $\chi$ denotes the Legendre chi function. where
Proof: █
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.
