Number Theory: An Introduction to Mathematics

5 Jacobian Elliptic Functions 527

K=K(τ):=πθ 002 ( 0 ;τ)/ 2 , K′=K′(τ):=τK(τ)/i, (51)

then we have

Poles ofsnu,cnu,dnu: u= 2 mK+( 2 n+ 1 )iK′(m,n∈Z). (52)

Zeros ofsnu: u= 2 mK+ 2 niK′,

cnu: u=( 2 m+ 1 )K+ 2 niK′,(m,n∈Z) (53)

dnu: u=( 2 m+ 1 )K+( 2 n+ 1 )iK′.

From the definitions (46) of the Jacobian elliptic functions and the behaviour of
the theta functions whenvis increased by 1 orτwe further obtain

snu=−sn(u+ 2 K)=sn(u+ 2 iK′),

cnu=−cn(u+ 2 K)=−cn(u+ 2 iK′),

dnu=dn(u+ 2 K)=−dn(u+ 2 iK′).

(54)

It follows that all three functions aredoubly-periodic. In fact snuhas periods 4Kand
2 iK′,cnuhas periods 4Kand 2K+ 2 iK′, and dnuhas periods 2Kand 4iK′. In each
case the ratio of the two periods is not real, sinceτ∈H.
Since any period must equal a difference between two poles, it must have the form
2 mK+ 2 niK′for somem,n∈Z.Since4Kand 2iK′are periods of snu,but2K
is not, and since any integral linear combination of periods is again a period, it fol-
lows that the periods of snuare precisely the integral linear combinations of 4Kand
2 iK′. Similarly the periods of cnuare the integral linear combinations of 4Kand
2 K+ 2 iK′, and the periods of dnuare the integral linear combinations of 2Kand
4 iK′.
It was shown in§3 that, if 0<λ<1, thenS(t,λ)has least positive period 2K(λ),
where

K(λ)=

∫ 1

0

dx/gλ(x)^1 /^2.

But, as we have seen, there is a unique pure imaginaryτ∈Hsuch thatλ=λ(τ),and
2 K[λ(τ)] is then the least positive period of sn^2 (u;τ). Since the periods of sn^2 (u;τ)
are 2mK+ 2 niK′(m,n∈Z), and sinceK,K′are real and positive whenτis pure
imaginary, it follows that

K[λ(τ)]=K(τ).

The domain of validity of this relation may be extended by appealing to results which
will be established in§6. In fact it holds, by analytic continuation, for allτin the region
Dillustrated in Figure 3, sinceλ(τ)∈Hforτ∈D.
From the definitions (46) of the Jacobian elliptic functions, the addition formulas
for the theta functions (Proposition 6) and the expression (48) forλ, we obtainaddition
formulasfor the Jacobian functions:

sn(u 1 +u 2 )=(snu 1 cnu 2 dnu 2 +snu 2 cnu 1 dnu 1 )/( 1 −λsn^2 u 1 sn^2 u 2 ),

cn(u 1 +u 2 )=(cnu 1 cnu 2 −snu 1 snu 2 dnu 1 dnu 2 )/( 1 −λsn^2 u 1 sn^2 u 2 ),

dn(u 1 +u 2 )=(dnu 1 dnu 2 −λsnu 1 snu 2 cnu 1 cnu 2 )/( 1 −λsn^2 u 1 sn^2 u 2 ).

(55)