Constant of Catalan

In mathematics, the constant of Catalan , named according to the mathematician Eugene Charles Catalan, is the number defined by:

$K\; =\; \backslash \; beta\; (2)\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \backslash \; infty\; \backslash \; frac\; \{(-\; 1)\; ^n\}\; \{(2n+1)\; ^2\}\; \backslash \; simeq$ 0,91596559417721901505…,

where $\backslash \; beta$ is the Fonction beta of Dirichlet. It is not known if constant K is rational or irrational and one expects that it is transcendent.

It is also equal to:

*$\{1\; \backslash \; over\; 2\}\; \backslash \; int\_\; \{0\}\; ^\; \{1\}\; F\; \backslash ,\; dk$ with $F\; =\; \backslash \; int\_\; \{0\}\; ^\; \{\backslash \; frac\; \{\backslash \; pi\}\; \{2\}\}\; \backslash \; frac\; \{\backslash ,\; D\; \backslash \; varphi\}\; \{\backslash \; sqrt\; \{1-k^2\; \backslash \; sin^2\; \backslash \; varphi\}\}$

*$\backslash \; int\_0^1\; \{\backslash \; arctan\; (U)\; \backslash \; over\; U\}\; \backslash ,\; du$

*$-\; \backslash \; int\_0^1\; \{\backslash \; ln\; (U)\; \backslash \; over\; 1+u^2\}\; \backslash ,\; du$

*$-\; \backslash \; int\_0^\; \{\backslash \; pi/4\}\; \{\backslash \; ln\; (\backslash \; tan\; (U))\; \backslash ,\; of\; the\}$

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