Lemniscate elliptic functions: Difference between revisions

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Undid revision 1306565853 by 208.114.63.4 (talk) unhelpful
Relation to the modular lambda function: use automatically sized parens around the large fraction in the second example; unfortunately this is still kind of ugly
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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( \frac\pi4 - {\arctan}\Bigglleft(\frac{2\sqrt[3]{2\sqrt{3}-2}-2\sqrt[3]{2-\sqrt{3}}+\sqrt{3}-1}{\sqrt[4]{12}}\Biggrright)\Biggrright)
\end{aligned}</math>