Parameter

A parameter $m$ used in Elliptic Integrals defined to be $m\equiv k^2$, where $k$ is the Modulus. An Elliptic Integral is written $I(\phi\vert m)$ when the parameter is used. The complementary parameter is defined by

$$m'\equiv 1-m,$$ (1)

where $m$ is the parameter. Let $q$ be the Nome, $k$ the Modulus, and $m\equiv k^2$ the Parameter. Then

$$q(m)=e^{-\pi K'(m)/K(m)}$$ (2)

where $K(m)$ is the complete Elliptic Integral of the First Kind. Then the inverse of $q(m)$ is given by

$$m(q)={{\vartheta_2}^4(q)\over{\vartheta_3}^4(q)},$$ (3)

where $\vartheta_i$ is a Theta Function.

See also Amplitude, Characteristic (Elliptic Integral), Elliptic Integral, Elliptic Integral of the First Kind, Modular Angle, Modulus (Elliptic Integral), Nome, Parameter, Theta Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 590, 1972.