526 XII Elliptic Functions
1 −λ(τ)=θ 014 ( 0 ;τ)/θ 004 ( 0 ;τ), (49)and from (39)–(40) that
cn^2 u= 1 −sn^2 u, dn^2 u= 1 −λsn^2 u. (50)Evidently (47) impliesd(sn^2 u)/du=2snucnudnu,
d^2 (sn^2 u)/du^2 = 2 (cn^2 udn^2 u−sn^2 udn^2 u−λsn^2 ucn^2 u).If we writeS(u)=S(u;τ):=sn^2 uand use (50), we can rewrite this in the form
d^2 S/du^2 =2[( 1 −S)( 1 −λS)−S( 1 −λS)−λS( 1 −S)]
= 6 λS^2 − 4 ( 1 +λ)S+ 2.SinceS( 0 )=S′( 0 )=0, we conclude thatS(u)coincides with the function denoted
by the same symbol in§3. However, it should be noted that nowλis not given, but is
determined byτ. Thus the question arises: can we chooseτ ∈H (the upper half-
plane) so thatλ(τ)is any prescribed complex number other than 0 or 1?
For many applications it is sufficient to know that we can chooseτ∈H so that
λ(τ)is any prescribed real number between 0 and 1. Since this case is much simpler,
we will deal with it now and defer treatmentof the general case until the next section.
We h av e
λ(τ)= 1 −θ 014 ( 0 ;τ)/θ 004 ( 0 ;τ)= 1 −∏∞
n= 1{( 1 −q^2 n−^1 )/( 1 +q^2 n−^1 )}^8 ,whereq=eπiτ.Ifτ=iy,wherey>0, then 0<q<1. Moreover, asyincreases
from 0 to∞,qdecreases from 1 to 0 and the infinite product increases from 0 to 1.
Thusλ(τ)decreases continuously from 1 to 0 and, for eachw ∈( 0 , 1 ),thereisa
unique pure imaginaryτ∈H such thatλ(τ)=w.
It should be mentioned that, also with our previous approach,S(u)could have
been recognized as the square of a meromorphic function by defining snu,cnu,dnu
to be the solution, forgivenλ∈C, of the system of differential equations (47) which
satisfies the initial condition sn 0=0, cn 0=dn 0=1.
Elliptic functions were first defined by Abel (1827) as the inverses of elliptic
integrals. His definitions were modified by Jacobi (1829) to accord with Legendre’s
normal form for elliptic integrals, and the functions snu,cnu,dnuare generally
known as theJacobian elliptic functions. The actual notation is due to Gudermann
(1838). The definition by means of theta functions was given later by Jacobi (1838) in
lectures.
Several properties of the Jacobian elliptic functions are easy consequences of the
later definition. In the first place, all three are meromorphic in the wholeu-plane, since
the theta functions are everywhere holomorphic. Their poles are determined by the
zeros ofθ 01 (v)and are all simple. Similarly, the zeros of snu,cnuand dnuare
determined by the zeros ofθ 11 (v),θ 10 (v)andθ 00 (v)respectively and are all simple. If
we put