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Elliptic Integral Singular Value k3

The third Singular Value $k_3$, corresponding to

\begin{displaymath}
K'(k_3)=\sqrt{3}\,K(k_3),
\end{displaymath} (1)

is given by
\begin{displaymath}
k_3=\sin\left({\pi\over 12}\right)={\textstyle{1\over 4}}(\sqrt{6}-\sqrt{2}).
\end{displaymath} (2)

As shown by Legendre,
\begin{displaymath}
K(k_3) = {\sqrt{\pi}\over 2\cdot 3^{3/4}} {\Gamma({1\over 6})\over \Gamma({2\over 3})}
\end{displaymath} (3)

(Whittaker and Watson 1990, p. 525). In addition,


\begin{displaymath}
E(k_3) ={\pi\over 4\sqrt{3}}{1\over K}+{\sqrt{3}+1\over 2\sq...
... 6})} + {2\Gamma({5\over 6})\over \Gamma({1\over 3})}}\right],
\end{displaymath} (4)

and
\begin{displaymath}
E'(k_3) ={\pi\sqrt{3}\over 4} {1\over K'(k_3)} + {\sqrt{3}-1\over 2\sqrt{3}} K'(k_3).
\end{displaymath} (5)

Summarizing,


$\displaystyle K[{\textstyle{1\over 4}}(\sqrt{6}-\sqrt{2})\,]$ $\textstyle =$ $\displaystyle {\sqrt{\pi}\over 2\cdot 3^{3/4}} {\Gamma({1\over 6})\over\Gamma({2\over 3})}$ (6)
$\displaystyle K'[{\textstyle{1\over 4}}(\sqrt{6}-\sqrt{2})\,]$ $\textstyle =$ $\displaystyle \sqrt{3}K ={\sqrt{\pi}\over 2\cdot 3^{1/4}} {\Gamma({1\over 6})\over \Gamma({2\over 3})}$ (7)
$\displaystyle E[{\textstyle{1\over 4}}(\sqrt{6}-\sqrt{2})\,]$ $\textstyle =$ $\displaystyle {1\over 4}\left({\pi\over \sqrt{3}}\right)^{1/2}\left[{\left({1+{...
...over\Gamma({5\over 6})} + {2\Gamma({5\over 6})\over \Gamma({1\over 3})}}\right]$ (8)
$\displaystyle E'[{\textstyle{1\over 4}}(\sqrt{6}-\sqrt{2})\,]$ $\textstyle =$ $\displaystyle {\sqrt{\pi}\over 2}\left[{3^{3/4}{\Gamma({\textstyle{2\over 3}})\...
...} {\Gamma({\textstyle{1\over 6}})\over \Gamma({\textstyle{2\over 3}})}}\right].$ (9)

(Whittaker and Watson 1990).

See also Theta Function


References

Ramanujan, S. ``Modular Equations and Approximations to $\pi$.'' Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 525-527 and 535, 1990.




© 1996-9 Eric W. Weisstein
1999-05-25