5 Jacobian Elliptic Functions 529Proposition 10Fo r a l l u∈Candτ∈H,sn(u;− 1 /τ)=−isn(iu;τ)/cn(iu;τ),
cn(u;− 1 /τ)= 1 /cn(iu;τ),
dn(u;− 1 /τ)=dn(iu;τ)/cn(iu;τ),Furthermore,
λ(− 1 /τ)= 1 −λ(τ),
K(− 1 /τ)=K′(τ).Proof Withv=u/πθ 002 ( 0 ;− 1 /τ)we have, by (38),sn(u;− 1 /τ)=−iθ 00 ( 0 ;− 1 /τ)θ 11 (v;− 1 /τ)/θ 10 ( 0 ;− 1 /τ)θ 01 (v;− 1 /τ)
=−θ 00 ( 0 ;τ)θ 11 (τv;τ)/θ 01 ( 0 ;τ)θ 10 (τv;τ).On the other hand, withv′=iu/πθ 002 ( 0 ;τ)we havesn(iu;τ)/cn(iu;τ)=−iθ 00 ( 0 ;τ)θ 11 (v′;τ)/θ 01 ( 0 ;τ)θ 10 (v′;τ).Sinceτv=v′, by comparing these two relations we obtain the first assertion of the
proposition.
The next two assertions may be obtained in the same way. The final two assertions
follow from (38), together with (48), (49) and (51). It follows from Proposition 10 that the evaluation of the Jacobian elliptic functions
for pure imaginary argument and parameterτmay be reduced to their evaluation for
real argument and parameter− 1 /τ.
From the definition (46) of the Jacobian elliptic functions and the duplication for-
mulas for the theta functions we can also obtain formulas for the Jacobian functions
when the parameterτis doubled (‘Landen’s transformation’):Proposition 11Fo r a l l u∈Candτ∈H,sn(u′′; 2 τ)=[1+( 1 −λ(τ))^1 /^2 ]sn(u;τ)cn(u;τ)/dn(u;τ),
cn(u′′; 2 τ)={ 1 −[1+( 1 −λ(τ))^1 /^2 ]sn^2 (u;τ)}/dn(u;τ),
dn(u′′; 2 τ)={ 1 −[1−( 1 −λ(τ))^1 /^2 ]sn^2 (u;τ)}/dn(u;τ),where u′′=[1+( 1 −λ(τ))^1 /^2 ]u and( 1 −λ(τ))^1 /^2 =θ 012 ( 0 ;τ)/θ 002 ( 0 ;τ).
Furthermore,λ( 2 τ)=λ^2 (τ)/[1+( 1 −λ(τ))^1 /^2 ]^4 ,
K( 2 τ)=[1+( 1 −λ(τ))^1 /^2 ]K(τ)/ 2.Proof Ifu=πθ 002 ( 0 ;τ)vandu′′=πθ 002 ( 0 ; 2 τ) 2 vthen, by Proposition 5,u′′= 2 θ 002 ( 0 ; 2 τ)u/θ 002 ( 0 ;τ)
=[θ 002 ( 0 ;τ)+θ^201 ( 0 ;τ)]u/θ 002 ( 0 ;τ).