# The Jacobian elliptic functions and the modular functionλ(τ)

• Komaravolu Chandrasekharan
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 281)

## Abstract

Letτbe a complex number, with Im τ>0. Letω1be defined as a function ofτby the relation
$${\omega _{1}} = \pi \theta _{3}^{2}(0,\tau ) = \pi {(1 + 1q + 2{q^{4}} + ...)^{2}},q = {e^{{\pi i\tau }}},\operatorname{Im} \tau >0$$
(1.1)
and let$${\omega _{2}} = {\omega _{1}} \cdot \tau$$. The Jacobian elliptic function snuis a doubly-periodic, meromorphic function ofu, with$$\left( {{\omega _{1}},{\omega _{2}}} \right)$$as a pair of basic periods, with two simple poles in each period-parallelogram, the sum of the residues at those poles being zero. It satisfies the differential equation
$${\left( {\frac{{dy}}{{du}}} \right)^{2}} = \left( {1 - {y^{2}}} \right)\left( {1 - {k^{2}}{y^{2}}} \right),\quad y = snu$$
(1.2)
, where
$${k^{2}} = {k^{2}}\left( \tau \right) = \frac{{\theta _{1}^{4}\left( {0,\tau } \right)}}{{\theta _{3}^{4}\left( {0,\tau } \right)}}$$
(1.3)
.