User:Exotic formulae

Alternating symmetric formula

${\frac {2^{3}}{\pi }}={\frac {\prod _{n=1}^{\infty }\left(1+{\frac {1}{4n^{2}-1}}\right)^{(-1)^{n+1}}}{\sum _{n=1}^{\infty }{\frac {1}{4n^{2}-1}}}}$

${\frac {1}{J_{4}\left(e^{-k\pi }\right)}}=\prod _{n=1}^{\infty }{\frac {e^{nk\pi }+1}{e^{nk\pi }-1}}$

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Inspired by your website

These are the formulas that I derived by visiting your website for the last fews months.

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Definition of:

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=1+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{4^{s}}}+\cdots$

$\eta (s)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n^{s}}}=1-{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}-{\frac {1}{4^{s}}}+\cdots$

$\lambda (s)=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{s}}}=1+{\frac {1}{3^{s}}}+{\frac {1}{5^{s}}}+{\frac {1}{7^{s}}}+\cdots$

$\beta (s)=\sum _{n=0}^{\infty }{\frac {(-1)^{k}}{(2n+1)^{s}}}=1-{\frac {1}{3^{s}}}+{\frac {1}{5^{s}}}-{\frac {1}{7^{s}}}+\cdots$

Where p is prime numbers

$\phi ={\frac {{\sqrt {5}}+1}{2}}$

See Mock theta function

$\theta (q),f(q),v(q),w(q)\cdots (1)$

$\theta ^{'}(q)={\sqrt {\frac {J_{2}(0,q)}{J_{3}(0,q)}}}$

$\psi (q)={\frac {J_{2}(0,q^{\frac {1}{2}})}{2q^{\frac {1}{8}}}}=\sum _{k=0}^{\infty }q^{\frac {k(k+1)}{2}}$

$\chi (q)=\prod _{k=1}^{\infty }\left(1+q^{2k-1}\right)$

$g_{n}-G_{n}\cdot$

Refer to Ramanujan g_n and G_n functions

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Derived Formulae

$\prod _{k=1}^{\infty }\left(1-{\frac {1}{e^{k\pi {\sqrt {n}}}}}\right)={\frac {2^{\frac {1}{2}}g_{n}^{2}\psi (e^{-\pi {\sqrt {n}}})}{e^{\frac {\pi {\sqrt {n}}}{12}}}}\cdots (1)$

$\prod _{k=1}^{\infty }\left(1-{\frac {1}{e^{2k\pi {\sqrt {n}}}}}\right)={\frac {2g_{n}^{2}G_{n}^{2}\psi (e^{-2\pi {\sqrt {n}}})}{e^{\frac {\pi {\sqrt {n}}}{6}}}}\cdots (2)$

$\prod _{k=1}^{\infty }\left(1+{\frac {1}{e^{k\pi {\sqrt {n}}}}}\right)={\frac {{\sqrt {2}}{\cdot }G_{n}^{2}{\cdot }\psi (e^{-\pi {\sqrt {n}}})}{e^{\frac {\pi {\sqrt {n}}}{12}}\cdot {J_{3}(0,e^{-\pi {\sqrt {n}}})}}}\cdots (4)$

$\psi (q^{-1})=\prod _{k=1}^{\infty }\left({\frac {q^{k}}{q^{k}-1}}\right)^{(-1)^{k+1}}\cdots (5)$

$2J_{1}{'}(0,q^{-{\frac {1}{2}}}){\cdot }{J_{1}{'}(0,q^{-1})}=\left[J_{2}(0,q^{-{\frac {1}{2}}}){\cdot }{J_{4}(0,q^{-1})}\right]^{3}\cdots (6)$

${\frac {J_{3}(0,e^{-\pi {\sqrt {n}}})}{J_{4}(0,e^{-\pi {\sqrt {n}}})}}={\frac {G_{n}^{2}}{g_{n}^{2}}}\cdots (7)$

$\prod _{k=1}^{\infty }{\frac {e^{k\pi {\sqrt {n}}}+1}{e^{k\pi {\sqrt {n}}}-1}}={\frac {G_{n}^{2}}{g_{n}^{2}\cdot {J_{3}(0,e^{-\pi {\sqrt {n}}})}}}\cdots (8)$

${\frac {\psi ^{2}(e^{-\pi {\sqrt {n}}})}{\psi (e^{-2\pi {\sqrt {n}}})}}=J_{3}(0,e^{-\pi {\sqrt {n}}})\cdots (9)$

${\frac {q^{\frac {1}{24}}\psi (q^{-1})\left[J_{1}^{'}(0,q^{-{\frac {1}{2}}})\right]^{\frac {1}{3}}}{2^{\frac {1}{3}}}}=\prod _{k=1}^{\infty }\left(1-{\frac {1}{q^{2k}}}\right)^{2}\cdots (10)$

${\frac {q^{\frac {1}{24}}\left[J_{1}^{'}(0,q^{-{\frac {1}{2}}})\right]^{\frac {1}{3}}}{2^{\frac {1}{3}}\psi (q^{-1})}}=\prod _{k=1}^{\infty }\left(1-{\frac {1}{q^{2k-1}}}\right)^{2}\cdots (11)$

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Jacobi theta function

$J_{2}(0,e^{-\pi })=J_{4}(0,e^{-\pi })=\left({\frac {\pi }{2}}\right)^{\frac {1}{4}}\cdot {\frac {1}{\Gamma \left({\frac {3}{4}}\right)}}\cdots (1)$

$J_{2}(0,e^{-{\frac {\pi }{2}}})={\frac {2^{\frac {5}{4}}}{e^{\frac {\pi }{6}}}}\cdots (2){error!}$

${\frac {J_{1}^{'}(0,e^{-\pi })}{J_{2}(0,e^{-\pi })}}=\left({\frac {\pi ^{2}}{2}}\right)^{\frac {1}{4}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}\cdots (3)$

$J_{1}^{'}(0,e^{-\pi })={\frac {(\pi )^{\frac {3}{4}}}{2^{\frac {1}{2}}}}\cdot {\frac {1}{\Gamma ^{3}\left({\frac {3}{4}}\right)}}\cdots (4)$

$J_{1}^{'}(0,e^{-2\pi })={\frac {(\pi )^{\frac {3}{4}}}{2^{\frac {13}{8}}}}\cdot {\frac {1}{\Gamma ^{3}\left({\frac {3}{4}}\right)}}\cdots (4b)$

$J_{1}^{'}(e^{-{\frac {\pi }{2}}})={\frac {\pi ^{\frac {3}{4}}}{2^{\frac {1}{8}}\cdot {\Gamma ^{3}\left({\frac {3}{4}}\right)}}}\cdots (5)$

$J_{4}\left(0,e^{-2\pi }\right)=\left({\frac {\pi ^{2}}{2}}\right)^{\frac {1}{8}}\cdot {\frac {1}{\Gamma \left({\frac {3}{4}}\right)}}\cdots (6)$

$\theta ^{'}\left(e^{-\pi }\right)={\frac {1}{2^{\frac {1}{8}}}}\cdots (7)$

$\psi (e^{-\pi })={\frac {\pi ^{\frac {1}{4}}\cdot {e^{\frac {\pi }{8}}}}{2^{\frac {5}{8}}\cdot {\Gamma \left({\frac {3}{4}}\right)}}}\cdots (8)$

$\psi (e^{-2\pi })=\left({\frac {e^{\pi }\cdot {\pi }}{2^{5}}}\right)^{\frac {1}{4}}{\frac {1}{\Gamma \left({\frac {3}{4}}\right)}}\cdots (9)$

$\chi (e^{-\pi })={\frac {2^{\frac {1}{4}}}{e^{\frac {\pi }{24}}}}\cdots (10)$

$2\theta (-e^{-\pi })-f(e^{-\pi })=\left({\frac {\pi ^{6}}{8e^{\pi }}}\right)^{\frac {1}{24}}{\frac {1}{\Gamma \left({\frac {3}{4}}\right)}}\cdots (11)$

$2\theta (-e^{-2\pi })-f(e^{-2\pi })=\left({\frac {8\pi ^{3}}{e^{\pi }}}\right)^{\frac {1}{12}}{\frac {1}{\Gamma \left({\frac {3}{4}}\right)}}\cdots (12)$

$v(e^{-\pi })+{e^{-\pi }}w(e^{-2\pi })={\frac {e^{\frac {\pi }{3}}\cdot {\pi ^{\frac {1}{4}}}}{2^{\frac {13}{8}}}}\cdot {\frac {1}{\Gamma \left({\frac {3}{4}}\right)}}\cdots (13)$

$v(e^{\frac {-\pi }{2}})+e^{\frac {-\pi }{2}}w(e^{-\pi })=2^{\frac {1}{8}}\cdots (14)$

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Infinite product

${\frac {8}{\pi }}={\frac {\prod _{n=1}^{\infty }\left(1+{\frac {1}{4n^{2}-1}}\right)^{(-1)^{n+1}}}{\sum _{n=1}^{\infty }{\frac {1}{4n^{2}-1}}}}\cdots (0)$

$\pi ={\frac {\prod _{n=1}^{\infty }\left[1+{\frac {1}{4(3n)^{2}-1}}\right]}{\sum _{n=1}^{\infty }{\frac {1}{4^{n}}}}}\cdots (1)$

${\frac {\Gamma ^{2}({\frac {1}{4}})}{\pi ^{\frac {3}{2}}}}={\frac {\prod _{n=1}^{\infty }\left[1+{\frac {1}{(4n-1)^{2}-1}}\right]}{\sum _{n=1}^{\infty }{\frac {1}{4n^{2}-1}}}}\cdots (2)$

${\frac {\Gamma ^{2}({\frac {1}{4}})}{\pi ^{\frac {5}{2}}}}={\sum _{n=1}^{\infty }{\frac {1}{4n^{2}-1}}}\cdot {{\prod _{n=1}^{\infty }\left[1+{\frac {1}{(4n-1)^{2}-1}}\right]^{2}}{\left[1+{\frac {1}{(4n+1)^{2}-1}}\right]}}\cdots (3)$

$\left[{\frac {\Gamma ^{(}{\frac {1}{4}})}{2}}\right]^{2}\cdot {\frac {1}{\sqrt {2\pi }}}={\frac {3}{2}}\cdot {\frac {4}{5}}\cdot {\frac {7}{6}}\cdot {\frac {8}{9}}\cdot {\frac {11}{10}}\cdot {\frac {12}{13}}=\prod _{k=2,4,6,...}^{\infty }\left({\frac {k+1}{k}}\right)^{(-1)^{{\frac {k}{2}}+1}}\cdots (4)$

${\frac {\left(2\pi \right)^{\frac {3}{2}}}{\Gamma ^{2}\left({\frac {1}{4}}\right)}}={\frac {4}{3}}\cdot {\frac {5}{6}}\cdot {\frac {8}{7}}\cdot {\frac {9}{10}}\cdots =\prod _{n=2}^{\infty }\left({\frac {2n-1}{2n}}\right)^{(-1)^{n}}\cdots (5)$

${\frac {4}{\pi }}=\lim _{n\to \infty }(4n+1)\cdot \left({\frac {1}{2}}\cdot {\frac {3}{4}}\cdot {\frac {5}{6}}\cdots {\frac {2n-1}{2n}}\right)^{2}\cdots (6)$

${\pi }=\lim _{n\to \infty }(4n+3)\cdot \left({\frac {2}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{7}}\cdots {\frac {2n}{2n+1}}\right)^{2}\cdots (7)$

${\frac {\pi }{2}}=\lim _{n\to \infty }{\sqrt {\frac {4n+3}{4n+1}}}\left({\frac {2^{2}}{2^{2}-1}}\cdot {\frac {4^{2}}{4^{2}-1}}\cdot {\frac {6^{2}}{6^{2}-1}}\cdots {\frac {4n^{2}}{4n^{2}-1}}\right)\cdots (8)$

$2^{\frac {1}{4}}=\prod _{n=1}^{\infty }{\frac {e^{(2n-1)\pi }+1}{e^{(3n-1)\pi }}}\cdot {\frac {(e^{n\pi }+1)^{3}}{e^{2n\pi }+1}}\cdots (9)$

$\pi =\lim _{n\to \infty }\left[\prod _{k=1}^{n}\left(1+{\frac {1}{2k-1}}\right)^{2}-\prod _{k=1}^{n-1}\left(1+{\frac {1}{2k-1}}\right)^{2}\right]\cdots (10)$

${\frac {4}{\pi }}=\lim _{n\to \infty }\left[\prod _{k=1}^{n}\left(1+{\frac {1}{2k}}\right)^{2}-\prod _{k=1}^{n-1}\left(1+{\frac {1}{2k}}\right)^{2}\right]\cdots (11)$

${\frac {e^{\pi }}{2^{3}}}=\prod _{n=1,3,5,...}^{\infty }\left({\frac {e^{n\pi }}{e^{n\pi }-1}}\right)^{24}\cdots (12)$

${\frac {e^{\pi }}{2^{6}}}=\prod _{n=1,3,5,...}^{\infty }\left({\frac {e^{n\pi }}{e^{n\pi }+1}}\right)^{24}\cdots (13)$

${\frac {e^{2\pi }}{2^{9}}}=\prod _{n=1,3,5,...}^{\infty }\left({\frac {e^{2n\pi }}{e^{2n\pi }-1}}\right)^{24}\cdots (14)$

$2=\prod _{n=1,3,5,...}^{\infty }\left({\frac {e^{n\pi }+1}{e^{n\pi }-1}}\right)^{8}\cdots (15)$

$\left({\frac {2}{\pi }}\right)^{\frac {1}{4}}\cdot {\Gamma \left({\frac {3}{4}}\right)}=\prod _{n=1}^{\infty }\left({\frac {e^{n\pi }+1}{e^{n\pi }-1}}\right)\cdots (16)$

$2^{\frac {1}{8}}=\prod _{n=1,3,5,...}^{\infty }\left({\frac {e^{n\pi }+1}{e^{n\pi }-1}}\right)\cdots (17)$

$\left({\frac {2}{\pi ^{2}}}\right)^{\frac {1}{8}}\cdot \Gamma \left({\frac {3}{4}}\right)=\prod _{n=1}^{\infty }{\frac {e^{2n\pi }+1}{e^{2n\pi }-1}}\cdots (18)$

$2=\prod _{n=1}^{\infty }\left[1+{\frac {1}{e^{(2n-1)\pi }-1}}\right]^{20}\left[\left(1+{\frac {1}{e^{n\pi }}}\right)\left(1-{\frac {1}{e^{n\pi }}}\right)^{2}\right]^{4}\cdots (19)$

$2=\prod _{n=1}^{\infty }\left(1+{\frac {1}{e^{n\pi }}}\right)^{20}\left(1-{\frac {1}{e^{2n\pi }+1}}\right)^{8}\left(1-{\frac {1}{e^{(2n-1)\pi }}}\right)^{4}\cdots (20)$

$2=\prod _{n=1}^{\infty }\left(1+{\frac {1}{e^{n\pi }}}\right)^{14}\left(1-{\frac {1}{e^{n\pi }}}\right)^{8}\left(1+{\frac {1}{e^{(2n-1)\pi }-1}}\right)^{10}\cdots (21)$

$e^{\pi }=\prod _{n=1}^{\infty }\left(1+{\frac {1}{e^{n\pi }-1}}\right)^{30}\left(1+{\frac {1}{e^{(2n-1)\pi }-1}}\right)^{42}\left(1-{\frac {1}{e^{2n\pi }}}\right)^{54}\cdots (22)$

${\frac {e^{\frac {\pi }{24}}}{2^{\frac {1}{8}}}}=\prod _{n=1}^{\infty }\left(1+{\frac {1}{e^{n\pi }}}\right)\cdots (23)$

${\frac {2^{\frac {1}{8}}}{e^{\frac {\pi }{24}}}}=\prod _{n=1}^{\infty }\left(1-{\frac {1}{e^{(2n-1)\pi }}}\right)\cdots (24)$

${\frac {2^{\frac {1}{4}}}{e^{\frac {\pi }{24}}}}=\prod _{n=1}^{\infty }\left(1+{\frac {1}{e^{(2n-1)\pi }}}\right)\cdots (25)$

${\frac {2^{\frac {3}{8}}}{e^{\frac {\pi }{12}}}}=\prod _{n=1}^{\infty }\left(1-{\frac {1}{e^{2(2n-1)\pi }}}\right)=\prod _{n=1}^{\infty }\left(1-{\frac {1}{e^{2n\pi }+1}}\right)\cdots (26)$

${\frac {e^{\frac {\pi }{12}}\cdot \pi ^{\frac {1}{4}}}{2^{\frac {1}{2}}\cdot \Gamma \left({\frac {3}{4}}\right)}}=\prod _{n=1}^{\infty }\left(1-{\frac {1}{e^{2n\pi }}}\right)\cdots (27)$

${\frac {\pi ^{\frac {1}{4}}\cdot {e^{\frac {\pi }{24}}}}{2^{\frac {3}{8}}\cdot {\Gamma \left({\frac {3}{4}}\right)}}}=\prod _{n=1}^{\infty }\left(1-{\frac {1}{e^{n\pi }}}\right)\cdots (28)$

${\frac {1}{2}}=\prod _{n=1}^{\infty }\left(1+{\frac {2}{n}}\right)^{(-1)^{n}}\cdots (29)$

${\frac {5}{\pi }}=\lim _{n\to \infty }{\frac {5n^{2}+15n+11}{(4n+7)(n+1)^{2}}}\left({\frac {3}{2}}\cdot {\frac {5}{4}}\cdot {\frac {7}{6}}\cdots {\frac {2n+1}{2n}}\right)^{2}\cdots (30)$

${\frac {6}{\pi }}=\lim _{n\to \infty }{\frac {6n^{2}+18n+13}{(4n+7)(n+1)^{2}}}\left({\frac {3}{2}}\cdot {\frac {5}{4}}\cdot {\frac {7}{6}}\cdots {\frac {2n+1}{2n}}\right)^{2}\cdots (31)$

${\frac {1}{\pi }}=\lim _{n\to \infty }{\frac {n+2}{(4n+7)(n+1)}}\left({\frac {3}{2}}\cdot {\frac {5}{4}}\cdot {\frac {7}{6}}\cdots {\frac {2n+1}{2n}}\right)^{2}\cdots (32)$

${\frac {\Gamma \left({\frac {1}{4}}\right)^{4}}{2^{5}}}=\lim _{n\to \infty }\left(8n+3\right)\left({\frac {4}{5}}\cdot {\frac {8}{9}}\cdot {\frac {12}{13}}\cdots {\frac {4n}{4n+1}}\right)^{4}\cdots (33)$

${\frac {2^{7}\pi ^{2}}{\Gamma ^{4}\left({\frac {1}{4}}\right)}}=\lim _{n\to \infty }\left(8n+5\right)\left({\frac {5}{6}}\cdot {\frac {9}{10}}\cdot {\frac {13}{14}}\cdots {\frac {4n+1}{4n+2}}\right)^{4}\cdots (34)$

$4\pi ^{2}=\lim _{n\to \infty }\left[(8n+5)(8n+3)\right]\left({\frac {2}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{7}}\cdots {\frac {2n}{2n+1}}\right)^{4}\cdots (35)$

${\frac {2}{\sqrt {\phi +2}}}={\frac {5^{2}}{5^{2}-1}}\cdot {\frac {15^{2}}{15^{2}-1}}\cdot {\frac {25^{2}}{25^{2}-1}}\cdot {\frac {35^{2}}{35^{2}-1}}\cdots =\prod _{n=0}^{\infty }\left[1+{\frac {1}{25(2n+1)^{2}-1}}\right]\cdots (36)$

${\frac {10}{{\phi }{\pi }{\sqrt {\phi +2}}}}={\frac {5^{2}}{5^{2}-1}}\cdot {\frac {10^{2}-1}{10^{2}}}\cdot {\frac {15^{2}}{15^{2}-1}}\cdot {\frac {20^{2}-1}{20^{2}}}\cdots =\prod _{n=1}^{\infty }\left(1+{\frac {1}{25n^{2}-1}}\right)^{(-1)^{n+1}}\cdots (37)$

${\frac {2{\sqrt {3}}}{\pi }}={\frac {3^{2}}{3^{2}-1}}\cdot {\frac {6^{2}-1}{6^{2}}}\cdot {\frac {9^{2}}{9^{2}-1}}\cdot {\frac {12^{2}-1}{12^{2}}}\cdots =\prod _{n=1}^{\infty }\left(1+{\frac {1}{9n^{2}-1}}\right)^{(-1)^{n+1}}\cdots (38)$

${\frac {\Gamma ^{4}\left({\frac {1}{4}}\right)}{8\pi ^{2}}}={\frac {3}{1}}\cdot {\frac {3}{5}}\cdot {\frac {7}{5}}\cdot {\frac {7}{9}}\cdot {\frac {11}{9}}\cdot {\frac {11}{13}}\cdots =\prod _{n=1}^{\infty }\left({\frac {2n+1}{2n-1}}\right)^{(-1)^{n+1}}\cdots (39)$

${\sqrt {2-{\sqrt {2}}}}={\frac {2}{3}}\cdot {\frac {6}{5}}\cdot {\frac {10}{11}}\cdots =\prod _{n=0}^{\infty }\left[1-{\frac {(-1)^{n}}{4n+2+(-1)^{n}}}\right]\cdots (40)$

$\left({\frac {\pi }{5}}\right)^{2}={\frac {\sum _{n=0}^{\infty }{\frac {(n!)^{2}}{\phi ^{2(n+1)}\cdot (2n+1)!}}}{\prod _{n=1}^{\infty }(1+{\frac {1}{25n^{2}-1}})^{(-1)^{n+1}}}}\cdots (41)$

${\frac {\pi }{2{\sqrt {3}}}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\right)^{2}\left({\frac {6n-4}{6n-1}}\right)\left({\frac {6n-2}{6n+1}}\right)\cdots (42)$

${\frac {\pi }{8\left({\sqrt {2}}-1\right)}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\right)^{2}\left({\frac {8n-5}{8n-1}}\right)\left({\frac {8n-3}{8n+1}}\right)\cdots (43)$

$\pi {\sqrt {2}}\cdot {\frac {\sqrt {5-{\sqrt {5}}}}{5\left({\sqrt {5}}-1\right)^{2}}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\right)^{2}\left({\frac {10n-6}{10n-1}}\right)\left({\frac {10n-4}{10n+1}}\right)\cdots (44)$

${\frac {\pi }{6\left({\sqrt {3}}-1\right)^{2}}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\right)^{2}\left({\frac {12n-7}{12n-1}}\right)\left({\frac {12n-5}{12n+1}}\right)\cdots (45)$

$\left({\frac {2}{e^{\frac {\pi }{4}}}}\right)^{\frac {1}{2}}=\prod _{n=1}^{\infty }{\frac {1}{\left(1+e^{-n\pi }\right)\left(1+e^{-2n\pi }\right)}}\cdots (46)$

$\left({\frac {2^{5}}{e^{\pi }}}\right)^{\frac {1}{8}}=\prod _{n=1}^{\infty }{\frac {1+e^{-n\pi }}{\left(1+e^{-2n\pi }\right)^{2}}}\cdots (47)$

$\left({\frac {2^{9}}{e^{2\pi }}}\right)^{\frac {1}{24}}=\prod _{n=1}^{\infty }{\frac {1}{1+e^{-2n\pi }}}\cdots (48)$

${\frac {1}{2^{\frac {1}{8}}}}=\prod _{n=1}^{\infty }{\frac {1+e^{-2n\pi }}{\left(1+e^{-n\pi }\right)^{2}}}\cdots (49)$

${\frac {2}{e^{\frac {5\pi }{24}}}}=\prod _{n=1}^{\infty }{\frac {1+e^{-n\pi }}{(1+e^{-2n\pi })^{3}}}\cdots (50)$

$\left({\frac {2}{e^{\frac {\pi }{6}}}}\right)^{\frac {1}{4}}=\prod _{n=1}^{\infty }{\frac {1+e^{-n\pi }}{1+e^{-2n\pi }}}\cdots (51)$

$\left({\frac {2^{21}}{e^{5\pi }}}\right)^{\frac {1}{24}}=\prod _{n=1}^{\infty }{\frac {1}{\left(1+e^{-n\pi }\right)\left(1+e^{-2n\pi }\right)^{2}}}\cdots (52)$

${\frac {5\phi }{\pi }}\left(\phi {\sqrt {2}}-{\sqrt {5-{\sqrt {5}}}}\right)^{2}={\frac {10^{2}}{10^{2}-1}}\cdot {\frac {20^{2}-1}{20^{2}}}\cdot {\frac {30^{2}}{30^{2}-1}}\cdot {\frac {40^{2}-1}{40^{2}}}\cdots (53)$

${\frac {2}{\psi ^{12}(e^{-\pi })}}=\prod _{n=1}^{\infty }\left({\frac {e^{n\pi }}{e^{n\pi }-1}}\right)^{4}\cdots (54)$

${\frac {e^{\pi }}{2^{3}\psi ^{12}(e^{-\pi })}}=\prod _{n=1}^{\infty }\left({\frac {e^{n\pi }}{e^{n\pi }-1}}\right)^{12}\cdots (55)$

${\frac {e^{\pi }\psi ^{12}(e^{-2\pi })}{\psi ^{42}(e^{-\pi })}}=\prod _{n=1}^{\infty }\left({\frac {e^{n\pi }}{e^{n\pi }-1}}\right)^{30}\cdots (56)$

${\frac {5^{4}\psi ^{26}(e^{-2\pi })\psi ^{7}(e^{-5\pi })}{\psi ^{91}(e^{-\pi })\psi ^{2}(e^{-10\pi })}}=\prod _{n=1}^{\infty }\left({\frac {e^{n\pi }}{e^{n\pi }-1}}\right)^{57}\left({\frac {e^{5n\pi }}{e^{5n\pi }-1}}\right)^{3}\cdots (57)$

${\frac {3^{3}\psi ^{14}(e^{-2\pi })\psi ^{7}(e^{-3\pi })}{\psi ^{49}(e^{-\pi })\psi ^{2}(e^{-6\pi })}}=\prod _{n=1}^{\infty }\left({\frac {e^{n\pi }}{e^{n\pi }-1}}\right)^{27}\left({\frac {e^{3n\pi }}{e^{3n\pi }-1}}\right)^{3}\cdots (58)$

$\left({\frac {e^{\pi }\cdot {\pi }}{2^{5}}}\right)^{\frac {1}{4}}\cdot {\frac {1}{\Gamma \left({\frac {3}{4}}\right)}}=\prod _{n=1}^{\infty }\left({\frac {e^{2n\pi }}{e^{2n\pi }-1}}\right)^{(-1)^{n+1}}\cdots (59)$

${\frac {\pi ^{\frac {1}{4}}\cdot {e^{\frac {\pi }{8}}}}{2^{\frac {5}{8}}\cdot {\Gamma \left({\frac {3}{4}}\right)}}}=\prod _{n=1}^{\infty }\left({\frac {e^{n\pi }}{e^{n\pi }-1}}\right)^{(-1)^{n+1}}\cdots (60)$

${\frac {e^{\frac {\pi }{8}}}{2^{\frac {5}{8}}}}=\prod _{n=1}^{\infty }\left({\frac {e^{n\pi }}{e^{n\pi }+1}}\right)^{(-1)^{n+1}}\cdots (61)$

${\frac {\pi ^{\frac {1}{4}}\cdot {e^{\frac {\pi }{6}}}}{2^{\frac {7}{8}}\cdot \Gamma \left({\frac {3}{4}}\right)}}=\prod _{n=1}^{\infty }\left(1-{\frac {1}{e^{4n\pi }}}\right)\cdots (62)$

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Infinite sum

$\pi =72\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n(e^{n\pi }-1)}}+24\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n(e^{2n\pi }-1)}}\cdots (1)$

${\frac {1}{2}}=\sum _{n=1}^{\infty }{\frac {1}{e^{\frac {(2n-1)^{2}\pi }{4}}}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{e^{n^{2}\pi }}}\cdots (2)$

${\frac {\pi }{4}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}\zeta (4n+2)}{(2n+1)\cdot {2^{2n+1}}}}\cdots (3)$

${\frac {\pi }{4}}=2\sum _{n=0}^{\infty }{\frac {\zeta (8n+2)}{(4n+1)\cdot 2^{4n+1}}}-\sum _{n=0}^{\infty }{\frac {\zeta (4n+2)}{(2n+1)\cdot 2^{2n+1}}}\cdots (4)$

${\frac {\pi }{4}}=\sum _{n=0}^{\infty }{\frac {\zeta (8n+2)}{(4n+1)\cdot 2^{4n+1}}}-\sum _{n=0}^{\infty }{\frac {\zeta (8n+6)}{(4n+3)\cdot 2^{4n+3}}}\cdots (5)$

$arctan\left({\frac {1}{2}}\right)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}\cdot 2^{2k+1}\left[\lambda (4k+2)-1\right]}{2k+1}}\cdots (6)$

$arctan\left({\frac {1}{2}}\right)=2\sum _{n=0}^{\infty }{\frac {2^{4n+1}\left[\lambda (8n+2)-1\right]}{4n+1}}-\sum _{n=0}^{\infty }{\frac {2^{2n+1}\left[\lambda (4n+2)-1\right]}{2n+1}}\cdots (7)$

$arctan\left({\frac {1}{2}}\right)=\sum _{n=0}^{\infty }{\frac {2^{4n+1}\left[\lambda (8n+2)-1\right]}{4n+1}}-\sum _{n=0}^{\infty }{\frac {2^{4n+3}\left[\lambda (8n+6)-1\right]}{4n+3}}\cdots (8)$

$\gamma =\sum _{n=1}^{\infty }{\frac {1-\beta (2n)}{n}}+\sum _{n=2}^{\infty }{\frac {(-1)^{n}\lambda (n)}{2^{n-2}\cdot {n}}}\cdots (9)$

${\frac {\pi ^{\frac {1}{4}}}{\Gamma \left({\frac {3}{4}}\right)}}=2\sum _{n=1}^{\infty }e^{\frac {-(2n-1)^{2}\pi }{4}}+4\sum _{n=1}^{\infty }e^{-(2n-1)^{2}\pi }\cdots (10)$

$2^{\frac {5}{4}}\sum _{n=1}^{\infty }e^{-{\frac {(2n-1)^{2}\pi }{4}}}=1+2\sum _{n=1}^{\infty }e^{-n^{2}\pi }\cdots (11)$

$2^{\frac {9}{8}}\sum _{n=1}^{\infty }e^{-{\frac {(2n-1)^{2}\pi }{4}}}=1+2\sum _{n=1}^{\infty }e^{-2n^{2}\pi }\cdots (12)$

$2^{\frac {1}{4}}\left(1+2\sum _{n=1}^{\infty }(-1)^{n}e^{-n^{2}\pi }\right)=1+2\sum _{n=1}^{\infty }e^{-n^{2}\pi }\cdots (13)$

$2^{\frac {1}{8}}\left(1+2\sum _{n=1}^{\infty }(-1)^{n}e^{-2n^{2}\pi }\right)=1+2\sum _{n=1}^{\infty }e^{-n^{2}\pi }\cdots (14)$

${\frac {\pi }{4}}=\sum _{k=1}^{\infty }\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}}{\left(2k^{2}+1\right)^{4n+3}}}\left[{\frac {(2k^{2}+1)^{2}}{4n+1}}+{\frac {2(2k^{2}+1)}{4n+2}}+{\frac {2}{4n+3}}\right]\cdots (15)$

${\frac {\pi }{4}}=\sum _{k=1}^{\infty }\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}}{\left(k^{2}+k\right)^{4n+3}}}\left[{\frac {(k^{2}+k)^{2}}{4n+1}}+{\frac {2(k^{2}+k)}{4n+2}}+{\frac {2}{4n+3}}\right]\cdots (16)$

$\pi =\sum _{n=0}^{\infty }{\frac {(-1)^{n}\cdot 2^{2n+4}}{\left(1+{\sqrt {3}}\right)^{4n+3}}}\left({\frac {2+{\sqrt {3}}}{4n+1}}+{\frac {1}{4n+3}}\right)\cdots (17)$

$1+{\frac {1}{e^{\pi }}}+{\frac {2}{e^{2\pi }}}+{\frac {3}{e^{3\pi }}}+{\frac {4}{e^{4\pi }}}+\cdots ={\frac {2^{\frac {3}{8}}\cdot {\Gamma \left({\frac {3}{4}}\right)}}{\pi ^{\frac {1}{4}}\cdot {e^{\frac {\pi }{24}}}}}\cdots (18)$

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Beta function

$ln\left[{\frac {\Gamma ^{2}({\frac {1}{4}})}{4{\sqrt {2\pi }}}}\right]=\sum _{n=1}^{\infty }{\frac {1-\beta (n)}{n}}\cdots (1)$

$ln\left[{\frac {\Gamma ^{4}({\frac {1}{4}})}{16\pi ^{2}}}\right]=\sum _{k=1}^{\infty }{\frac {1-\beta (2k)}{k}}\cdots (2)$

$ln{\sqrt {\frac {\pi }{2}}}=\sum _{k=0}^{\infty }{\frac {1-\beta (2k+1)}{2k+1}}\cdots (3)$

$ln\left[{\frac {4\pi ^{2}}{\sqrt {2\pi }}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {1}{4}}\right)}}\right]=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}\left[1-\beta (n)\right]}{n}}\cdots (4)$

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prime number

$ln{\sqrt {\frac {7}{6}}}=\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\sum _{p=2}^{\infty }{\frac {1}{p^{8k+4}}}\cdots (1)$

$ln{\sqrt {\frac {5}{2}}}=\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\sum _{p=2}^{\infty }{\frac {1}{p^{4k+2}}}\cdots (2)$

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Zeta function

$ln\left({\frac {\pi }{sinh\pi }}\right)=\sum _{n=1}^{\infty }{\frac {(-1)^{k}\zeta (2n)}{n}}\cdots (1)$

$ln{\sqrt {\frac {3}{2}}}=\sum _{k=0}^{\infty }{\frac {\zeta (6k+3)-1}{2k+1}}\cdots (2)$

$ln{\sqrt {\frac {sinh\pi }{\pi }}}=\sum _{k=0}^{\infty }{\frac {\zeta (4k+2)-1}{2k+1}}\cdots (3)$

$ln\pi -\gamma =\sum _{k=2}^{\infty }{\frac {\zeta (k)}{k\cdot 2^{k-1}}}\cdots (4)$

$ln2-\gamma =\sum _{k=1}^{\infty }{\frac {\zeta (2k+1)}{(2k+1)2^{2k}}}\cdots (5)$

$ln(2\pi )=\sum _{n=1}^{\infty }{\frac {\left(2^{2n+1}+1\right)\zeta (2n)-2^{2n+1}}{2^{2n}\cdot {n}}}\cdots (6)$

$ln\left({\frac {\pi }{2}}\right)=\sum _{n=1}^{\infty }{\frac {\zeta (2n)}{2^{2n}n}}\cdots (7)$

$ln\left({\frac {\pi }{3}}\right)=\sum _{n=1}^{\infty }{\frac {\zeta (2n)}{6^{2n}n}}\cdots (8)$

$ln\left({\frac {\pi \phi }{5}}\right)=\sum _{n=1}^{\infty }{\frac {\zeta (2n)}{10^{2n}n}}\cdots (9)$

$ln\left({\frac {2\pi \phi }{5{\sqrt {\phi +2}}}}\right)=\sum _{n=1}^{\infty }{\frac {\zeta (2n)}{5^{2n}n}}\cdots (10)$

$ln(2)=\sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}\cdots (11)$

$ln\left({\frac {4\pi }{sinh\pi }}\right)=\sum _{n=1}^{\infty }{\frac {\zeta (4n)-1}{n}}\cdots (12)$

${\gamma }=\lim _{k\to \infty }\left[\sum _{n=1}^{k}{{\frac {\zeta (2n)}{n}}-ln2k}\right]\cdots (13)$

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Eta function

$ln\left[{\frac {\Gamma ^{2}\left({\frac {1}{4}}\right)}{4{\sqrt {2\pi }}}}\right]=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}\eta (k)}{2^{k}\cdot {k}}}\cdots (1)$

$ln\left({\frac {2}{\pi }}\right)=\sum _{n=1}^{\infty }{\frac {(-1)^{n}\eta (n)}{n}}\cdots (2)$

$ln{\sqrt {2}}=\sum _{k=0}^{\infty }{\frac {1-\eta (2k+1)}{2k+1}}\cdots (3)$

$ln\left({\frac {\pi }{e}}\right)=2\sum _{n=0}^{\infty }{\frac {1-\eta (2n)}{2n+1}}\cdots (4)$

$ln\left({\frac {8}{\pi ^{2}}}\right)=\sum _{n=1}^{\infty }{\frac {\eta (2n)-1}{n}}\cdots (5)$

$ln\left({\frac {2}{\pi }}\right)=\sum _{n=1}^{\infty }{\frac {\eta (n)-1}{n}}\cdots (6)$

$ln\left({\frac {10}{{\pi }{\phi }{\sqrt {\phi +2}}}}\right)=\sum _{n=1}^{\infty }{\frac {\eta (2n)}{5^{2n}\cdot {n}}}\cdots (7)$

$ln\left({\frac {4}{\pi }}\right)=\sum _{n=1}^{\infty }{\frac {\eta (2n)}{2^{2n}\cdot {n}}}\cdots (8)$

$ln\left({\frac {2{\sqrt {3}}}{\pi }}\right)=\sum _{n=1}^{\infty }{\frac {\eta (2n)}{3^{2n}\cdot {n}}}\cdots (9)$

$ln\left[{\frac {\Gamma ^{2}\left({\frac {1}{4}}\right)}{2\pi {\sqrt {2}}}}\right]=\sum _{n=0}^{\infty }{\frac {\eta (2n+1)}{2^{2n+1}\cdot {(2n+1)}}}\cdots (10)$

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Lambda function

$ln\left({\frac {4}{\pi }}\right)=\sum _{n=1}^{\infty }{\frac {\lambda (2n)-1}{n}}\cdots (1)$

$ln\left({\frac {1}{\pi }}\right)+2-\gamma =2\sum _{n=2}^{\infty }{\frac {\lambda (n)-1}{n}}\cdots (2)$

${\gamma }=\lim _{k\to \infty }\left[\sum _{n=1}^{k}{{\frac {\lambda (2n)}{m}}-ln\left({\frac {4k}{\pi }}\right)}\right]\cdots (3)$

$\gamma +ln\left({\frac {4}{\pi }}\right)=2\sum _{n=2}^{\infty }{\frac {(-1)^{n}\lambda (n)}{n}}\cdots (4)$

$ln\left[{\frac {16\pi ^{2}}{\Gamma ^{4}\left({\frac {1}{4}}\right)}}\right]+\gamma =4\sum _{n=2}^{\infty }{\frac {(-1)^{n}\lambda (n)}{2^{n}\cdot {n}}}\cdots (5)$

$ln\left({\frac {2}{\sqrt {\phi +2}}}\right)=\sum _{n=1}^{\infty }{\frac {\lambda (2n)}{5^{2n}\cdot {n}}}\cdots (6)$

$ln{\sqrt {2}}=\sum _{n=1}^{\infty }{\frac {\lambda (2n)}{2^{2n}\cdot {n}}}\cdots (7)$

$ln\left[{\sqrt {2(2-{\sqrt {2}})}}\right]=\sum _{n=1}^{\infty }{\frac {\lambda (2n)}{4^{2n}\cdot {n}}}\cdots (8)$

$ln\left({\frac {2}{\sqrt {3}}}\right)=\sum _{n=1}^{\infty }{\frac {\lambda (2n)}{3^{2n}\cdot {n}}}\cdots (9)$

$ln\left[{\frac {\Gamma ^{2}\left({\frac {1}{4}}\right)}{2\pi {\sqrt {2}}}}\right]-{\frac {\gamma }{2}}=2\sum _{n=1}^{\infty }{\frac {\lambda (2n+1)}{2^{2n+1}\cdot {(2n+1)}}}\cdots (10)$

$ln\left(\phi ^{2}{\sqrt {2}}-\phi {\sqrt {5-{\sqrt {5}}}}\right)=\sum _{n=1}^{\infty }{\frac {\lambda (2n)}{n\cdot 10^{2n}}}\cdots (11)$

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$\pi +ln\left({\frac {1}{2^{3}}}\right)=24\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n(e^{n\pi }-1)}}\cdots (1)$

$2\pi +ln\left({\frac {1}{2^{9}}}\right)=24\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n(e^{2n\pi }-1)}}\cdots (2)$

$ln2=24\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n(e^{n\pi }-1)}}-8\sum _{n=1}^{\infty }{\frac {1}{n(e^{n\pi }-1)}}\cdots (3)$

${4\pi }+ln\left({\frac {1}{2^{18}}}\right)=24\sum _{k=1}^{\infty }{\frac {1}{k}}\left({\frac {1}{e^{2k\pi }+1}}+{\frac {1}{e^{2k\pi }-1}}\right)\cdots (4)$

${\frac {1}{8}}ln2=\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {1}{e^{(2k+1)\pi }+1}}+{\frac {1}{e^{(2k+1)\pi }-1}}\right)\cdots (5)$

${\pi }+ln\left({\frac {1}{2^{6}}}\right)=12\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n}}\left({\frac {1}{e^{n\pi }+1}}+{\frac {1}{e^{n\pi }-1}}\right)\cdots (6)$

$ln{\frac {\pi ^{\frac {1}{4}}}{\Gamma ({\frac {3}{4}})}}=2\sum _{k=0}^{\infty }{\frac {1}{(2k+1)(e^{\pi (2k+1)}+1)}}\cdots (7)$

$ln\left[\left({\frac {2}{\pi }}\right)^{\frac {1}{4}}\cdot {\Gamma \left({\frac {3}{4}}\right)}\right]=2\sum _{k=0}^{\infty }{\frac {1}{(2k+1)(e^{\pi (2k+1)}-1)}}\cdots (8)$

$ln\left[\Gamma \left({\frac {3}{4}}\right)\cdot \left({\frac {2}{\pi ^{2}}}\right)^{\frac {1}{8}}\right]=\sum _{n=1}^{\infty }{\frac {1}{2k-1}}\left({\frac {1}{e^{(2n-1)\pi }-1}}-{\frac {1}{e^{(2n-1)\pi }+1}}\right)\cdots (9)$

$ln\left[\left({\frac {2}{\pi ^{2}}}\right)^{\frac {1}{8}}\cdot {\Gamma \left({\frac {3}{4}}\right)}\right]=2\sum _{n=0}^{\infty }{\frac {1}{2n+1}}\left({\frac {1}{e^{2\pi (2n+1)}-1}}\right)\cdots (10)$

$ln3=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{3n}}\left({\frac {84}{e^{n\pi }-1}}-{\frac {28}{e^{2n\pi }-1}}-{\frac {12}{e^{3n\pi }-1}}+{\frac {4}{e^{6n\pi }-1}}\right)-\sum _{n=1}^{\infty }{\frac {1}{3n}}\left({\frac {8}{e^{n\pi }-1}}-{\frac {8}{e^{3n\pi }-1}}\right)\cdots (11)$

$ln5=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{4n}}\left({\frac {156}{e^{n\pi }-1}}-{\frac {52}{e^{2n\pi }-1}}-{\frac {19}{e^{5n\pi }-1}}+{\frac {4}{e^{10n\pi }-1}}\right)-\sum _{n=1}^{\infty }{\frac {1}{4n}}\left({\frac {8}{e^{n\pi }}}-{\frac {1}{e^{5n\pi }-1}}\right)\cdots (12)$

$ln\left({\sqrt {\phi +2}}-\phi \right)+{\frac {2\pi }{5}}=\sum _{n=1}^{\infty }{\frac {e^{2n\pi }-e^{4n\pi }-e^{6n\pi }+e^{8n\pi }}{n(1-e^{10n\pi })}}\cdots (13)$

$ln\left({\frac {2^{\frac {1}{2}}}{e^{\frac {\pi }{12}}}}\right)=\sum _{m=1}^{\infty }{\frac {1-tanh\left({\frac {m\pi }{2}}\right)}{m}}\cdots (14)$

$ln\left[\left({\frac {e^{\pi }\cdot {\pi }}{2^{5}}}\right)^{\frac {1}{2}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}\right]=\sum _{m=1}^{\infty }{\frac {1-tanh(m\pi )}{m}}\cdots (15)$

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x > 1

$ln2=\sum _{n=1}^{\infty }{\frac {1}{(2^{x}-x-1)n}}\sum _{r=1}^{2^{x}-2}{\frac {2^{x}-r-1}{(r+1)^{2n}}}\cdots (1)$

$ln3=\sum _{k=0}^{\infty }{\frac {1}{2^{2k}\cdot (2k+1)}}\cdots (2)$

$ln2=\sum _{n=1}^{\infty }{\frac {1}{n}}\left({\frac {1}{2^{2n}}}+{\frac {2}{3^{2n}}}+{\frac {2}{4^{2n}}}+{\frac {1}{5^{2n}}}\right)\cdots (3)$

$ln\left({\frac {2{\sqrt {2}}}{e}}\right)=\sum _{n=1}^{\infty }{\frac {1}{n}}\left[{\frac {1}{4}}\left({\frac {1}{3^{2n}}}\right)+{\frac {1}{8}}\left({\frac {1}{5^{2n}}}+{\frac {1}{7^{2n}}}\right)+{\frac {1}{16}}\left({\frac {1}{9^{2n}}}+{\frac {1}{11^{2n}}}+{\frac {1}{13^{2n}}}+{\frac {1}{15^{2n}}}\right)+\cdots \right]\cdots (4)$

$\sum _{k=1}^{\infty }{\frac {\zeta (2k)\left[(2^{2k}-1)\zeta (2k)-2^{2k}\right]}{k}}+ln\left({\frac {\pi }{2}}\right)=\sum _{n=1}^{\infty }{\frac {1}{n}}\sum _{r=1}^{\infty }\sum _{a=3}^{\infty }\left[{\frac {1}{(2ar-a-1)^{2n}}}+{\frac {2}{(2ar-a)^{2n}}}+{\frac {1}{(2ar-a+1)^{2n}}}\right]\cdots (5)$

Where K = 8.700... is the constant from polygon circumscribing.

$ln\left({\frac {2K}{\pi }}\right)=\sum _{n=1}^{\infty }{\frac {1}{n}}\sum _{r=1}^{\infty }\sum _{k=1}^{\infty }\left[{\frac {1}{(2rk+r-1)^{2n}}}+{\frac {2}{(2rk+r)^{2n}}}+{\frac {1}{(2rk+r+1)^{2n}}}\right]\cdots (6)$

$ln\left[\sum _{n=0}^{\infty }{\frac {(n!)^{2}}{\phi ^{2n}(2n+1)!}}\right]=\sum _{n=1}^{\infty }{\frac {\zeta (2n)}{5^{2n}\cdot {n}}}\cdots (7)$

$\sum _{n=1}^{\infty }{\frac {1-\beta (n)}{n}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}\eta (n)}{2^{n}\cdot {n}}}\cdots (8)$

$ln\left({\frac {\pi }{\sqrt {6}}}\right)=\sum _{n=1}^{\infty }{\frac {\zeta (2n)}{n}}\left[{\frac {\left({\frac {3^{n}+1}{2}}\right)^{2}+\left({\frac {3^{n}-1}{2}}\right)^{2}}{(6^{n})^{2}}}\right]\cdots (9)$

$ln\left[{\frac {\Gamma \left({\frac {1}{4}}\right)}{\pi }}\right]+{\frac {\gamma }{4}}=\sum _{n=2}^{\infty }{\frac {(-1)^{n}\lambda (n)}{n}}\left(1-{\frac {1}{2^{n}}}\right)\cdots (10)$

$ln\left[{\frac {4}{\Gamma \left({\frac {1}{4}}\right)}}\right]+{\frac {3\gamma }{4}}=\sum _{n=2}^{\infty }{\frac {(-1)^{n}\lambda (n)}{n}}\left(1+{\frac {1}{2^{n}}}\right)\cdots (11)$

$ln\left({\frac {\pi ^{2}}{8}}\cdot {\sqrt {\frac {\pi }{e}}}\right)=\sum _{n=1}^{\infty }{\frac {(3n+1)}{n(2n+1)}}\cdot \left[1-\eta (2n)\right]\cdots (12)$

$ln\left({\frac {8}{\pi ^{2}}}\cdot {\sqrt {\frac {\pi }{e}}}\right)=\sum _{n=1}^{\infty }{\frac {(n+1)}{n(2n+1)}}\cdot \left[\eta (2n)-1\right]\cdots (13)$

$ln\left({\frac {4}{e}}\right)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n(n+1)}}\cdots (14)$

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Other

$\pi =2arctan\left({\frac {1}{cos^{2}\phi }}\right)+arctan\left({\frac {2sin^{6}\phi +2sin^{4}\phi -4}{sin^{8}\phi -2sin^{4}\phi -4sin^{2}\phi }}\right)\cdots (1)$

${\frac {tan^{6}\phi -tan^{4}\phi +2}{tan^{6}\phi -2tan^{2}\phi +4}}\cdot {cos^{2}\phi }={\frac {sin^{6}\phi +sin^{4}\phi -2}{sin^{6}\phi -2sin^{2}\phi -4}}\cdots (2)$

${\frac {cot^{6}\phi -cot^{4}\phi +2}{cot^{6}\phi -2cot^{2}\phi +4}}\cdot {sin^{2}\phi }={\frac {cos^{6}\phi +cos^{4}\phi -2}{cos^{6}\phi -2cos^{2}\phi -4}}\cdots (3)$