Elliptic Integral Singular Value

When the Modulus $k$ has a singular value, the complete elliptic integrals may be computed in analytic form in terms of Gamma Functions. Abel (quoted in Whittaker and Watson 1990, p. 525) proved that whenever

$${K'(k)\over K(k)}={a+b\sqrt{n}\over c+d\sqrt{n}},$$ (1)

where $a$, $b$, $c$, $d$, and $n$ are Integers, $K(k)$ is a complete Elliptic Integral of the First Kind, and $K'(k_)\equiv K(\sqrt{1-k^2}\,)$ is the complementary complete Elliptic Integral of the First Kind, then the Modulus $k$ is the Root of an algebraic equation with Integer Coefficients.

A Modulus $k_r$ such that

$${K'(k_r)\over K(k_r)}=\sqrt{r},$$ (2)

is called a singular value of the elliptic integral. The Elliptic Lambda Function $\lambda^*(r)$ gives the value of $k_r$. Selberg and Chowla (1967) showed that $K(\lambda^*(r))$ and $E(\lambda^*(r))$ are expressible in terms of a finite number of Gamma Functions. The complete Elliptic Integrals of the Second Kind $E(k_r)$ and $E'(k_r)$ can be expressed in terms of $K(k_r)$ and $K'(k_r)$ with the aid of the Elliptic Alpha Function $\alpha(r)$.

The following table gives the values of $K(k_r)$ for small integral $r$ in terms of Gamma Functions.

$$\begin{eqnarray*} K(k_1)&=&{\Gamma^2({\textstyle{1\over 4}})\over 4\sqrt{\pi}}\... ...5}+2\over 20} {\Gamma^2({\textstyle{1\over 4}})\over\sqrt{\pi}}, \end{eqnarray*}$$

where $\Gamma(z)$ is the Gamma Function and $C_1$ is an algebraic number (Borwein and Borwein 1987, p. 298).

Borwein and Zucker (1992) give amazing expressions for singular values of complete elliptic integrals in terms of Central Beta Functions

$$\beta(p)\equiv B(p,p).$$ (3)

Furthermore, they show that $K(k_n)$ is always expressible in terms of these functions for $n\equiv 1,2\ \left({{\rm mod\ } {4}}\right)$. In such cases, the $\Gamma$ functions appearing in the expression are of the form $\Gamma(t/4n)$ where $1\leq t\leq (2n-1)$ and $(t,4n)=1$. The terms in the numerator depend on the sign of the Kronecker Symbol $\{t/4n\}$. Values for the first few $n$ are

$K(k_1)=2^{-2}\beta({\textstyle{1\over 4}})$
$K(k_2)=2^{-13/4}\beta({\textstyle{1\over 8}})$
$K(k_3)=2^{-4/3}3^{-1/4}\beta({\textstyle{1\over 3}})=2^{-5/3}3^{-3/4}\beta({\textstyle{1\over 6}})$
$K(k_5)=2^{-33/20}5^{-5/8}(11+5\sqrt{5}\,)^{1/4}\sin({\textstyle{1\over 20}}\pi)\beta({\textstyle{1\over 2}})$
$\qquad =2^{-29/20}5^{-3/8}(1+\sqrt{5}\,)^{1/4}\sin({\textstyle{3\over 20}}\pi)\beta({\textstyle{3\over 20}})$
$K(k_6)=2^{-47/12}3^{-3/4}(\sqrt{2}-1)(\sqrt{3}+1)\beta({\textstyle{1\over 24}})$
$\qquad =2^{-43/12}3^{-1/4}(\sqrt{3}-1)\beta({\textstyle{5\over 24}})$
$K(k_7)=2\cdot 7^{-3/4}\sin({\textstyle{1\over 7}}\pi)\sin({\textstyle{2\over 7}}\pi)B({\textstyle{1\over 7}}, {\textstyle{2\over 7}})$
$\qquad =2^{-2/7}7^{-1/4}{\beta({\textstyle{1\over 7}})\beta({\textstyle{2\over 7}})\over\beta({\textstyle{1\over 14}})}$
$K(k_{10})=2^{-61/20}5^{-1/4}(\sqrt{5}-2)^{1/2}(\sqrt{10}+3){\beta({\textstyle{1\over 8}})\beta({\textstyle{7\over 40}})\over\beta({\textstyle{1\over 3}}{40})}$
$\qquad =2^{-15/4}5^{-3/4}(\sqrt{5}-2)^{1/2}{\beta({\textstyle{1\over 40}})\beta({\textstyle{1\over 9}}{40})\over\beta({\textstyle{3\over 8}})}$
$K(k_{11})=R\cdot 2^{-7/11}\sin({\textstyle{1\over 11}}\pi)\sin({\textstyle{3\over 11}}\pi)B({\textstyle{1\over 22}}, {\textstyle{3\over 22}})$
$K(k_{13})=2^{-3}13^{-5/8}(5\sqrt{13}+18)^{1/4}[\tan({\textstyle{1\over 52}}\pi)... ...1\over 52}})\beta({\textstyle{9\over 52}})\over\beta({\textstyle{23\over 52}})}$
$K(k_{14})=\sqrt{\sqrt{4\sqrt{2}+2}+\sqrt{2}+\sqrt{2\sqrt{2}-1}}\cdot 2^{-13/4}7... ...3\over 56}})\beta({\textstyle{1\over 8}})\over \beta({\textstyle{11\over 56}})}$
$K(k_{15})=2^{-1}3^{-3/4}5^{-7/12}B({\textstyle{1\over 15}}, {\textstyle{4\over 15}})$
$\qquad ={2^{-2}3^{-3/4}5^{-3/4}(\sqrt{5}-1)\beta({\textstyle{1\over 15}})\beta({\textstyle{4\over 15}})\over\beta({\textstyle{1\over 3}})}$
$K(k_{17})=C_2\left[{\beta({\textstyle{1\over 68}})\beta({\textstyle{3\over 68}}... ...ver\beta({\textstyle{5\over 68}})\beta({\textstyle{15\over 68}})}\right]^{1/4},$

where $R$ is the Real Root of

$$x^3-4x=4=0$$ (4)

and $C_2$ is an algebraic number (Borwein and Zucker 1992). Note that $K(k_{11})$ is the only value in the above list which cannot be expressed in terms of Central Beta Functions.

Using the Elliptic Alpha Function, the Elliptic Integrals of the Second Kind can also be found from

$$E = {\pi\over 4\sqrt{r}K}+\left[{1-{\alpha(r)\over\sqrt{r}}}\right]K$$ (5)

$$E' = {\pi\over 4K}+\alpha(r)K,$$ (6)

and by definition,

$$K'=K\sqrt{n}.$$ (7)

See also Central Beta Function, Elliptic Alpha Function, Elliptic Delta Function, Elliptic Integral of the First Kind, Elliptic Integral of the Second Kind, Elliptic Lambda Function, Gamma Function, Modulus (Elliptic Integral)

References

Abel, N. H. ``Recherches sur les fonctions elliptiques.'' J. reine angew. Math. 3, 160-190, 1828. Reprinted in Abel, N. H. Oeuvres Completes (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., p. 377, 1988.

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.

Borwein, J. M. and Zucker, I. J. ``Elliptic Integral Evaluation of the Gamma Function at Rational Values of Small Denominator.'' IMA J. Numerical Analysis 12, 519-526, 1992.

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

Glasser, M. L. and Wood, V. E. ``A Closed Form Evaluation of the Elliptic Integral.'' Math. Comput. 22, 535-536, 1971.

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

Weisstein, E. W. ``Elliptic Singular Values.'' Mathematica notebook EllipticSingular.m.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 524-528, 1990.

Wrigge, S. ``An Elliptic Integral Identity.'' Math. Comput. 27, 837-840, 1973.

Zucker, I. J. ``The Evaluation in Terms of $\Gamma$-Functions of the Periods of Elliptic Curves Admitting Complex Multiplication.'' Math. Proc. Cambridge Phil. Soc. 82, 111-118, 1977.