Singular Value

A Modulus $k_r$ such that

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

where $K(k)$ is a complete Elliptic Integral of the First Kind, and $K'(k_r)\equiv K(\sqrt{1-{k_r}^2}\,)$. The Elliptic Lambda Function $\lambda^*(r)$ gives the value of $k_r$.

Abel (quoted in Whittaker and Watson 1990, p. 525) proved that if $r$ is an Integer, or more generally whenever

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

where $a$, $b$, $c$, $d$, and $n$ are Integers, then the Modulus $k$ is the Root of an algebraic equation with Integer Coefficients.

See also Elliptic Integral Singular Value, Elliptic Integral of the First Kind, Elliptic Lambda Function, Modulus (Elliptic Integral)

References

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