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Modular Equation

The modular equation of degree $n$ gives an algebraic connection of the form

{K'(l)\over K(l)}=n {K'(k)\over K(k)}
\end{displaymath} (1)

between the Transcendental Complete Elliptic Integrals of the First Kind with moduli $k$ and $l$. 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} (2)

exists, and $M$ is called the Modular Function Multiplier. In general, if $p$ is an Odd Prime, then the modular equation is given by
\end{displaymath} (3)

u_p\equiv (-1)^{(p^2-1)/8}[\lambda(q^p)]^{1/8}\equiv (-1)^{(p^2-1)/8}u(q^p),
\end{displaymath} (4)

$\lambda$ is a Elliptic Lambda Function, and
q\equiv e^{i\pi t}
\end{displaymath} (5)

(Borwein and Borwein 1987, p. 126). An Elliptic Integral identity gives
{K'(k)\over K(k)}=2 {K'\left({2\sqrt{k}\over 1+k}\right)\over K\left({2\sqrt{k}\over 1+k}\right)},
\end{displaymath} (6)

so the modular equation of degree 2 is
l={2\sqrt{k}\over 1+k}\,
\end{displaymath} (7)

which can be written as
\end{displaymath} (8)

A few low order modular equations written in terms of $k$ and $l$ are
$\quad \Omega_2=l^2(1+k)^2-4k=0$ (9)
$\quad \Omega_7=(kl)^{1/4}+(k'l')^{1/4}-1=0$ (10)
$\quad \Omega_{23}=(kl)^{1/4}+(k'l')^{1/4}+2^{2/3}(klk'l')^{1/12}-1=0.$

In terms of $u$ and $v$,
$\displaystyle \Omega_3(u,v)$ $\textstyle =$ $\displaystyle u^4-v^4+2uv(1-u^2v^2)=0$ (12)
$\displaystyle \Omega_5(u,v)$ $\textstyle =$ $\displaystyle v^6-u^6+5u^2v^2(v^2-u^2)+4uv(u^4v^4-1)$  
  $\textstyle =$ $\displaystyle \left({u\over v}\right)^3+\left({v\over u}\right)^3=2\left({u^2v^2-{1\over u^2v^2}}\right)=0$  

\end{displaymath} (14)

u^2\equiv \sqrt{k}={\vartheta _2(q)\over \vartheta_3(q)}
\end{displaymath} (15)

v^2\equiv \sqrt{l}={\vartheta _2(q^p)\over \vartheta_3(q^p)}.
\end{displaymath} (16)

Here, $\vartheta_i$ are Theta Functions.

A modular equation of degree $2^r$ for $r\geq 2$ can be obtained by iterating the equation for $2^{r-1}$. Modular equations for Prime $p$ from 3 to 23 are given in Borwein and Borwein (1987).

Quadratic modular identities include

{\vartheta _3(q)\over \vartheta _3(q^4)}-1=\left[{{{\vartheta _3}^2(q^2)\over {\vartheta _3}^2(q^4)}-1}\right]^{1/2}.
\end{displaymath} (17)

Cubic identities include
\left[{3 {\vartheta _2(q^9)\over \vartheta _2(q)}-1}\right]^3=9{{\vartheta _2}^4(q^3)\over {\vartheta _2}^4(q)}-1
\end{displaymath} (18)

\left[{3 {\vartheta _3(q^9)\over \vartheta _3(q)}-1}\right]^3=9{{\vartheta _3}^4(q^3)\over {\vartheta _3}^4(q)}-1
\end{displaymath} (19)

\left[{3 {\vartheta _4(q^9)\over \vartheta _4(q)}-1}\right]^3=9{{\vartheta _4}^4(q^3)\over {\vartheta _4}^4(q)}-1.
\end{displaymath} (20)

A seventh-order identity is
\sqrt{\vartheta _3(q)\vartheta _3(q^7)}-\sqrt{\vartheta _4(q)\vartheta _4(q^7)}=\sqrt{\vartheta _2(q)\vartheta _2(q^7)}.
\end{displaymath} (21)

From Ramanujan (1913-1914),
\end{displaymath} (22)

\end{displaymath} (23)

See also Schläfli's Modular Form


Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 127-132, 1987.

Hanna, M. ``The Modular Equations.'' Proc. London Math. Soc. 28, 46-52, 1928.

Ramanujan, S. ``Modular Equations and Approximations to $\pi$.'' Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.

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© 1996-9 Eric W. Weisstein