Elliptic Lambda Function

The $\lambda$ Group is the Subgroup of the Gamma Group with $a$ and $d$ Odd; $b$ and $c$ Even. The function

$$\lambda(t)\equiv \lambda(q)\equiv k^2(q) = \left[{\vartheta _2(q)\over \vartheta _3(q)}\right]^4,$$ (1)

where

$$q\equiv e^{i\pi t}$$ (2)

is a $\lambda$-Modular Function and $\vartheta _i$ are Theta Functions.

$\lambda^*(r)$ gives the value of the Modulus $k_r$ for which the complementary and normal complete Elliptic Integrals of the First Kind are related by

$${K'(k_r)\over K(k_r)}=\sqrt{r}.$$ (3)

It can be computed from

$$\lambda^*(r)\equiv k(q)={{\vartheta_2}^2(q)\over{\vartheta_3}^2(q)},$$ (4)

where

$$q\equiv e^{-\pi\sqrt{r}},$$ (5)

and $\vartheta _i$ is a Theta Function.

From the definition of the lambda function,

$$\lambda^*(r')=\lambda^*\left({1\over r}\right)={\lambda^*}'(r).$$ (6)

For all rational $r$, $K(\lambda^*(r))$ and $E(\lambda^*(r))$ are expressible in terms of a finite number of Gamma Functions (Selberg and Chowla 1967). $\lambda^*(r)$ is related to the Ramanujan g- and G-Functions by

$$\lambda^*(n) = {\textstyle{1\over 2}}(\sqrt{1+G_n^{-12}}-\sqrt{1-G_n^{-12}}\,)$$ (7)

$$\lambda^*(n) = g_n^6(\sqrt{g_n^{12}+g_n^{-12}}-g_n^6).$$ (8)

Special values are

$\lambda^*({\textstyle{2\over 29}})=(13\sqrt{58}-99)(\sqrt{2}+1)^6$
$\lambda^*({\textstyle{2\over 5}})=(\sqrt{10}-3)(\sqrt{2}+1)^2$
$\lambda^*({\textstyle{2\over 3}})=(2-\sqrt{3}\,)(\sqrt{2}+\sqrt{3}\,)$
$\lambda^*({\textstyle{3\over 4}})=(\sqrt{3}-\sqrt{2}\,)^2(\sqrt{2}+1)^2$
$\lambda^*(1)={1\over\sqrt{2}}$
$\lambda^*(2)=\sqrt{2}-1$
$\lambda^*(3)={\textstyle{1\over 4}}\sqrt{2}(\sqrt{3}-1)$
$\lambda^*(4)=3-2\sqrt{2}$
$\lambda^*(5)={\textstyle{1\over 2}}\left({\sqrt{\sqrt{5}-1}-\sqrt{3-\sqrt{5}}\,}\right)$
$\lambda^*(6)=(2-\sqrt{3}\,)(\sqrt{3}-\sqrt{2}\,)$
$\lambda^*(7)={\textstyle{1\over 8}}\sqrt{2}(3-\sqrt{7})$
$\lambda^*(8)=\left({\sqrt{2}+1-\sqrt{2\sqrt{2}+2}\,}\right)^2$
$\lambda^*(9)={\textstyle{1\over 2}}(\sqrt{2}-3^{1/4})(\sqrt{3}-1)$
$\lambda^*(10)=(\sqrt{10}-3)(\sqrt{2}-1)^2$

$\lambda^*(11)={\textstyle{1\over 12}}\sqrt{6}\left({\sqrt{1+2x_{11}-4{x_{11}}^{-1}}-\sqrt{11+2x_{11}-4{x_{11}}^{-1}}\,}\right)$
$\lambda^*(12)=(\sqrt{3}-\sqrt{2}\,)^2(\sqrt{2}-1)^2 = 15-10\sqrt{2}+8\sqrt{3}-6\sqrt{6}$
$\lambda^*(13)={\textstyle{1\over 2}}(\sqrt{5\sqrt{13}-17}-\sqrt{19-5\sqrt{13}}\,)$
$\lambda^*(14)=-11-8\sqrt{2}-2(\sqrt{2}+2)\sqrt{5+4\sqrt{2}\,}$
$\qquad +\sqrt{11+8\sqrt{2}}(2+2\sqrt{2}+\sqrt{2}\sqrt{5+4\sqrt{2}}\,)$
$\lambda^*(15)={\textstyle{1\over 16}}\sqrt{2}(3-\sqrt{5}\,)(\sqrt{5}-\sqrt{3}\,)(2-\sqrt{3}\,)$
$\lambda^*(16)={(2^{1/4}-1)^2\over (2^{1/4}+1)^2}$
$\lambda^*(17)={\textstyle{1\over 4}}\sqrt{2}(\sqrt{42+10\sqrt{17}}-13\sqrt{-3+\sqrt{17}}\sqrt{5+\sqrt{17}}$
$\qquad -3\sqrt{17}\sqrt{-3+\sqrt{17}}\sqrt{5+\sqrt{17}}$
$\qquad -\sqrt{-38-10\sqrt{17}+13\sqrt{-3+\sqrt{17}}}\sqrt{5+\sqrt{17}}+3\sqrt{17}\sqrt{-3+\sqrt{17}}\sqrt{5+\sqrt{17}})$
$\lambda^*(18)=(\sqrt{2}-1)^3(2-\sqrt{3}\,)^2$
$\lambda^*(22)=(3\sqrt{11}-7\sqrt{2}\,)(10-3\sqrt{11}\,)$
$\lambda^*(30)=(\sqrt{3}-\sqrt{2}\,)^2(2-\sqrt{3}\,)(\sqrt{6}-\sqrt{5}\,)(4-\sqrt{15}\,)$
$\lambda^*(34)=(\sqrt{2}-1)^2(3\sqrt{2}-\sqrt{17}\,)(\sqrt{297+72\sqrt{17}}-\sqrt{296+72\sqrt{17}})$
$\lambda^*(42)=(\sqrt{2}-1)^2(2-\sqrt{3}\,)^2(\sqrt{7}-\sqrt{6}\,)(8-3\sqrt{7}\,)$
$\lambda^*(58)=(13\sqrt{58}-99)(\sqrt{2}-1)^6$
$\lambda^*(210)=(\sqrt{2}-1)^2(2-\sqrt{3})(\sqrt{7}-\sqrt{6})^2(8-3\sqrt{7})$
$\qquad \times(\sqrt{10}-3)^2(4-\sqrt{15})^2(\sqrt{15}-\sqrt{14})(6-\sqrt{35}),$

where

$$x_{11}\equiv (17+3\sqrt{33}\,)^{1/3}.$$

In addition,

$$\lambda^*(1') = {1\over\sqrt{2}}$$

$$\lambda^*(2') = \sqrt{2\sqrt{2}-2}$$

$$\lambda^*(3') = {\textstyle{1\over 4}}\sqrt{2}(\sqrt{3}+1)$$

$$\lambda^*(4') = 2^{1/4}(2\sqrt{2}-2)$$

$$\lambda^*(5') = {\textstyle{1\over 2}}\left({\sqrt{\sqrt{5}-1}+\sqrt{3-\sqrt{5}}\,}\right)$$

$$\lambda^*(7') = {\textstyle{1\over 8}}\sqrt{2}\,(3+\sqrt{7})$$

$$\lambda^*(9') = {\textstyle{1\over 2}}(\sqrt{2}+3^{1/4})(\sqrt{3}-1)$$

$$\lambda^*(12') = 2\sqrt{-208+147\sqrt{2}-120\sqrt{3}+85\sqrt{6}}.$$

See also Elliptic Alpha Function, Elliptic Integral of the First Kind, Modulus (Elliptic Integral), Ramanujan g- and G-Functions, Theta Function

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 139 and 298, 1987.

Bowman, F. Introduction to Elliptic Functions, with Applications. New York: Dover, pp. 75, 95, and 98, 1961.

Selberg, A. and Chowla, S. ``On Epstein's Zeta-Function.'' J. Reine. Angew. Math. 227, 86-110, 1967.

Watson, G. N. ``Some Singular Moduli (1).'' Quart. J. Math. 3, 81-98, 1932.