Lemniscate elliptic functions: Difference between revisions

Revision as of 01:34, 17 August 2025

Latest revision as of 18:12, 18 August 2025

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:<math>

\operatorname{cl}^2 \tfrac12x = \frac{1+\operatorname{cl}x \sqrt{1+\operatorname{sl}^2x}}{1 + \sqrt{1+\operatorname{sl}^2x}+1}

</math>

:<math>

\operatorname{sl}^2 \tfrac12x = \frac{1-\operatorname{cl}x\sqrt{1+\operatorname{sl}^2x}}{1 + \sqrt{1+\operatorname{sl}^2x}+1}

</math>

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

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

= {\tan}\Bigglleft(\vphantom{\frac\Big|\Big|}\right. \frac\pi4 - {\arctan}\Bigglleft(\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}}\Biggrleft.\left.\vphantom{\frac\Big|\Big|}\right)\Biggrright)

\end{aligned}</math>