Modular Equation

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

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

(1)

between the Transcendental Complete Elliptic Integrals of the First Kind with moduli

and

. When

and

satisfy a modular equation, a relationship of the form

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

(2)

exists, and

is called the Modular Function Multiplier. In general, if

is an Odd Prime, then the modular equation is given by

$\begin{displaymath} \Omega_p(u,v)=(v-u_0)(v-u_1)\cdots(v-u_p), \end{displaymath}$

(3)

where

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

(4)

$\lambda$ is a Elliptic Lambda Function, and

$\begin{displaymath} q\equiv e^{i\pi t} \end{displaymath}$

(5)

(Borwein and Borwein 1987, p. 126). An Elliptic Integral identity gives

$\begin{displaymath} {K'(k)\over K(k)}=2 {K'\left({2\sqrt{k}\over 1+k}\right)\over K\left({2\sqrt{k}\over 1+k}\right)}, \end{displaymath}$

(6)

so the modular equation of degree 2 is

$\begin{displaymath} l={2\sqrt{k}\over 1+k}\, \end{displaymath}$

(7)

which can be written as

$\begin{displaymath} l^2(1+k)^2=4k. \end{displaymath}$

(8)

A few low order modular equations written in terms of

and

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)
$\quad \Omega_{23}=(kl)^{1/4}+(k'l')^{1/4}+2^{2/3}(klk'l')^{1/12}-1=0.$
	(11)

In terms of

and

$\displaystyle \Omega_3(u,v)$	$\textstyle =$	$\displaystyle u^4-v^4+2uv(1-u^2v^2)=0$	(12)
$\displaystyle \Omega_5(u,v)$	$\textstyle =$	$\displaystyle v^6-u^6+5u^2v^2(v^2-u^2)+4uv(u^4v^4-1)$
	$\textstyle =$	$\displaystyle \left({u\over v}\right)^3+\left({v\over u}\right)^3=2\left({u^2v^2-{1\over u^2v^2}}\right)=0$
			(13)

$\begin{displaymath} \Omega_7(u,v)=(1-u^8)(1-v^8)-(1-uv)^8=0, \end{displaymath}$

(14)

where

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

(15)

and

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

(16)

Here, $\vartheta_i$ are Theta Functions.

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

Quadratic modular identities include

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

(17)

Cubic identities include

$\begin{displaymath} \left[{3 {\vartheta _2(q^9)\over \vartheta _2(q)}-1}\right]^3=9{{\vartheta _2}^4(q^3)\over {\vartheta _2}^4(q)}-1 \end{displaymath}$

(18)

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

(19)

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

(20)

A seventh-order identity is

$\begin{displaymath} \sqrt{\vartheta _3(q)\vartheta _3(q^7)}-\sqrt{\vartheta _4(q)\vartheta _4(q^7)}=\sqrt{\vartheta _2(q)\vartheta _2(q^7)}. \end{displaymath}$

(21)

From Ramanujan (1913-1914),

$\begin{displaymath} (1+q)(1+q^3)(1+q^5)\cdots=2^{1/6}q^{1/24}(kk')^{-1/12} \end{displaymath}$

(22)

$\begin{displaymath} (1-q)(1-q^3)(1-q^5)\cdots=2^{1/6}q^{1/24}k^{-1/12}k'^{1/6}. \end{displaymath}$

(23)

See also Schläfli's Modular Form

References

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.