512 XII Elliptic Functions
since
S(u)^2 S′(v)^2 −S′(u)^2 S(v)^2 =S(u)^2 gλ[S(v)]−S(v)^2 gλ[S(u)]
= 4 S(u)S(v)[S(u)−S(v)][1−λS(u)S(v)].Replacingvby−vin (11) and subtracting the result from (11), we obtain
S(u+v)−S(u−v)=S′(u)S′(v)/[1−λS(u)S(v)]^2. (12)In particular, forv=u,
S( 2 u)=gλ[S(u)]/[1−λS^2 (u)]^2. (13)We recall that a function ismeromorphicin a connected open setDif it is holomor-
phic throughoutD, except for isolated singularities which are poles. Since, by (13),
S( 2 t)is a rational function ofS(t), it follows that ifS(t)is meromorphic and a
solution (wherever it is finite) of the differential equation (7) in an open disc|t|<R,
then its definition can be extended so that it is meromorphic and a solution (wherever it
is finite) of the differential equation (7) also in the disc|t|< 2 R. But the fundamental
existence and uniqueness theorem guarantees thatS(t)is holomorphic in a neighbour-
hood of the origin. Consequently we can extend its definition so that it is meromorphic
and a solution of (7) in the whole complex planeC.
Further properties of the functionS(t)may be derived from the differential equa-
tion (7). For any constantsα,β,ify(t)=αS(βt),theny( 0 )=y′( 0 )=0. It is readily
seen thaty(t)satisfies a differential equation of the form (7) if and only if eitherα=1,
β=±1orα=λ,λβ^2 =1, and in the latter case withλreplaced by 1/λin (7). It
follows that, for anyλ=0,
S(t, 1 /λ)=λS(λ−^1 /^2 t,λ). (14)By differentiation it may be shown also thatS(it,λ)/[S(it,λ)−1], wherei^2 =−1,
is a solution of the differential equation (7) withλreplaced by 1−λ. It follows that
S(t, 1 −λ)=S(it,λ)/[S(it,λ)−1]. (15)By combining (14) and (15) we obtain, for anyλ= 0 ,1, three more relations:S(t, 1 /( 1 −λ))=( 1 −λ)S(i( 1 −λ)−^1 /^2 t,λ)/[S(i( 1 −λ)−^1 /^2 t,λ)−1], (16)
S(t,(λ− 1 )/λ)=λS(iλ−^1 /^2 t,λ)/[λS(iλ−^1 /^2 t,λ)−1], (17)
S(t,λ/(λ− 1 ))=( 1 −λ)S(( 1 −λ)−^1 /^2 t,λ)/[1−λS(( 1 −λ)−^1 /^2 t,λ)]. (18)As in§1, it follows from (14)–(18) that the evaluation ofS(t,λ)for allt,λ∈C
reduces to its evaluation forλin the region|λ− 1 |≤1, 0≤Rλ≤ 1 /2. Similarly
it follows from (14) and (18) that the evaluation ofS(t,λ)for allt,λ∈Rreduces to
its evaluation forλin the interval 0<λ<1. We now show thatS(t,λ)can then be
calculated by theAGMalgorithm.
It is easily verified that if
z(t)=( 1 +√
λ)^2 S(t,λ)/[1+√
λS(t,λ)]^2 ,