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Modular Function Multiplier

When $k$ and $l$ satisfy a Modular Equation, a relationship of the form

{M(l,k)\,dy\over \sqrt{(1-y^2)(1-l^2y^2)}} = {dx\over \sqrt{(1-x^2)(1-k^2x^2)}}
\end{displaymath} (1)

exists, and $M$ is called the multiplier. The multiplier of degree $n$ can be given by
M_n(l,k)\equiv {{\vartheta _3}^2(q)\over {\vartheta _3}^2(q^{1/p})} = {K(k)\over K(l)},
\end{displaymath} (2)

where $\vartheta_i$ is a Theta Function and $K(k)$ is a complete Elliptic Integral of the First Kind.

The first few multipliers in terms of $l$ and $k$ are

$\displaystyle M_2(l,k)$ $\textstyle =$ $\displaystyle {1\over 1+k}={1+l'\over 2}$ (3)
$\displaystyle M_3(l,k)$ $\textstyle =$ $\displaystyle {1-\sqrt{l^3\over k}\over 1-\sqrt{k^3\over l}}.$ (4)

In terms of the $u$ and $v$ defined for Modular Equations,
$\displaystyle M_3$ $\textstyle =$ $\displaystyle {v\over v+2u^3}={2v^3-u\over 3u}$ (5)
$\displaystyle M_5$ $\textstyle =$ $\displaystyle {v(1-uv^3)\over v-u^5}={u+v^5\over 5u(1+u^3v)}$ (6)
$\displaystyle M_7$ $\textstyle =$ $\displaystyle {v(1-uv)[1-uv+(uv)^2]\over v-u^7}$  
  $\textstyle =$ $\displaystyle {v^7-u\over 7u(1-uv)[1-uv+(uv)^2]}.$ (7)

© 1996-9 Eric W. Weisstein