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}](m_1435.gif) |
(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}](m_1436.gif) |
(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}](m_1437.gif) |
(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}](m_1438.gif) |
(4) |
is a Elliptic Lambda Function, and
![\begin{displaymath}
q\equiv e^{i\pi t}
\end{displaymath}](m_1439.gif) |
(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}](m_1440.gif) |
(6) |
so the modular equation of degree 2 is
![\begin{displaymath}
l={2\sqrt{k}\over 1+k}\,
\end{displaymath}](m_1441.gif) |
(7) |
which can be written as
![\begin{displaymath}
l^2(1+k)^2=4k.
\end{displaymath}](m_1442.gif) |
(8) |
A few low order modular equations written in terms of
and
are
In terms of
and
,
![\begin{displaymath}
\Omega_7(u,v)=(1-u^8)(1-v^8)-(1-uv)^8=0,
\end{displaymath}](m_1451.gif) |
(14) |
where
![\begin{displaymath}
u^2\equiv \sqrt{k}={\vartheta _2(q)\over \vartheta_3(q)}
\end{displaymath}](m_1452.gif) |
(15) |
and
![\begin{displaymath}
v^2\equiv \sqrt{l}={\vartheta _2(q^p)\over \vartheta_3(q^p)}.
\end{displaymath}](m_1453.gif) |
(16) |
Here,
are Theta Functions.
A modular equation of degree
for
can be obtained by iterating the equation for
. 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}](m_1458.gif) |
(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}](m_1459.gif) |
(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}](m_1460.gif) |
(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}](m_1461.gif) |
(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}](m_1462.gif) |
(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}](m_1463.gif) |
(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}](m_1464.gif) |
(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
.'' Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.
© 1996-9 Eric W. Weisstein
1999-05-26