Number Theory: An Introduction to Mathematics

5 Jacobian Elliptic Functions 525

Proposition 8Fo r a l lv∈Candτ∈H,

{θ 00 (v)/θ 01 (v)}′=πiθ 102 ( 0 )θ 10 (v)θ 11 (v)/θ 012 (v), (42)

{θ 10 (v)/θ 01 (v)}′=πiθ 002 ( 0 )θ 00 (v)θ 11 (v)/θ 012 (v), (43)

{θ 11 (v)/θ 01 (v)}′=πiθ 012 ( 0 )θ 00 (v)θ 10 (v)/θ 012 (v), (44)

{θ 01 ′(v)/θ 01 (v)}′=θ 01 ′′( 0 )/θ 01 ( 0 )+π^2 θ 002 ( 0 )θ 102 ( 0 )θ 112 (v)/θ 012 (v). (45)

Proof By differentiating the second relation of Proposition 6 with respect towand
then puttingw=0, we obtain

θ 00 ( 0 )θ 01 ( 0 )[θ 00 ′(v)θ 01 (v)−θ 00 (v)θ 01 ′(v)]=θ 10 ( 0 )θ 11 ′( 0 )θ 10 (v)θ 11 (v),

since not onlyθ 11 ( 0 )=0butalsoθ 00 ′( 0 )=θ 01 ′( 0 )=θ 10 ′( 0 )=0. Dividing byθ 012 (v)
and recalling the expression (36) forθ 11 ′( 0 ), we obtain (42). Similarly, from the third
and fourth relations of Proposition 6 we obtain (43) and (44).
In the same way, if we differentiate the first relation of Proposition 6 twice with
respect towand then putw=0, we obtain

θ 012 ( 0 )[θ 01 ′′(v)θ 01 (v)−θ 01 ′(v)^2 ]=θ 01 ( 0 )θ 01 ′′( 0 )θ 012 (v)−θ 11 ′( 0 )^2 θ 112 (v).

Hence, using (36) again, we obtain (45).

We are now in a position to make the connection between theta functions and
elliptic functions.

5 Jacobian Elliptic Functions...................................

The behaviour of the theta functions when their argument is increased by 1 orτmakes
it clear that doubly-periodic functions may be constructed from their quotients. We put

snu=sn(u;τ):=−iθ 00 ( 0 )θ 11 (v)/θ 10 ( 0 )θ 01 (v),

cnu=cn(u;τ):=θ 01 ( 0 )θ 10 (v)/θ 10 ( 0 )θ 01 (v),

dnu=dn(u;τ):=θ 01 ( 0 )θ 00 (v)/θ 00 ( 0 )θ 01 (v),

(46)

whereu=πθ 002 ( 0 )v.
The constant multiples are chosen so that, in addition to sn 0 = 0, we have
cn 0=dn 0=1. The independent variable is scaled so that, by (42)–(44),

d(snu)/du=cnudnu,

d(cnu)/du=−snudnu,

d(dnu)/du=−λsnucnu,

(47)

where
λ=λ(τ):=θ 104 ( 0 ;τ)/θ 004 ( 0 ;τ). (48)
It follows at once from the definitions that snuis an odd function ofu, whereas
cnuand dnuare even functions ofu. It follows from (41) that