Lemniscate elliptic functions: Difference between revisions

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:<math>\begin{aligned}

&{\operatorname{sl}}\leftbigl(\tfrac1{14}\varpi\rightbigr)\,{\operatorname{sl}}\leftbigl(\tfrac3{14}\varpi\rightbigr)\,{\operatorname{sl}}\leftbigl(\tfrac5{14}\varpi\rightbigr) \\[7mu]

&\quad {}= \sqrt[8]{\frac{\lambda (7i)}{1-\lambda (7i)}}

= {\tan}\leftBigl({\tfrac{1}{2}\arccsc}\leftBigl(\tfrac{1}{2}\sqrt{8\sqrt{7}+21}+\tfrac{1}{2}\sqrt{7}+1\rightBigr)\rightBigr)

\\[7mu]

&\quad {}= \frac 2 {2 + \sqrt{7} + \sqrt{21 + 8 \sqrt{7}} + \sqrt{2 {14 + 6 \sqrt{7} + \sqrt{455 + 172 \sqrt{7}}}}}

\\[18mu]

& {\operatorname{sl}}\leftbigl(\tfrac1{18}\varpi\rightbigr)\, {\operatorname{sl}}\leftbigl(\tfrac3{18}\varpi\rightbigr)\,{\operatorname{sl}}\leftbigl(\tfrac5{18}\varpi\rightbigr)\,{\operatorname{sl}}\leftbigl(\tfrac7{18}\varpi\rightbigr) \\[-3mu]

&\quad {}= \sqrt[8]{\frac{\lambda (9i)}{1-\lambda (9i)}}

= {\tan}\left(\vphantom{\frac\Big|\Big|}\right. \frac\pi4 - {\arctan}\left(\vphantom{\frac\Big|\Big|}\right.\frac{2\sqrt[3]{2\sqrt{3}-2}-2\sqrt[3]{2-\sqrt{3}}+\sqrt{3}-1}{\sqrt[4]{12}}\left.\left.\vphantom{\frac\Big|\Big|}\right)\right)

\end{aligned}</math>