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22 Jacobian Elliptic FunctionsProperties

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

With t and

22.12.1 τ=iK(k)/K(k),
22.12.2 2Kksn(2Kt,k)=n=-πsin(π(t-(n+12)τ))=n=-(m=-(-1)mt-m-(n+12)τ),
22.12.3 2iKkcn(2Kt,k)=n=-(-1)nπsin(π(t-(n+12)τ))=n=-(m=-(-1)m+nt-m-(n+12)τ),
22.12.4 2iKdn(2Kt,k)=limNn=-NN(-1)nπtan(π(t-(n+12)τ))=limNn=-NN(-1)n(limMm=-MM1t-m-(n+12)τ).

The double sums in (22.12.2)–(22.12.4) are convergent but not absolutely convergent, hence the order of the summations is important. Compare §20.5(iii).

22.12.5 2Kkcd(2Kt,k) =n=-πsin(π(t+12-(n+12)τ))
=n=-(m=-(-1)mt+12-m-(n+12)τ),
22.12.6 -2iKkksd(2Kt,k) =n=-(-1)nπsin(π(t+12-(n+12)τ))
=n=-(m=-(-1)m+nt+12-m-(n+12)τ),
22.12.7 2iKknd(2Kt,k) =limNn=-NN(-1)nπtan(π(t+12-(n+12)τ))
=limNn=-NN(-1)nlimM(m=-MM1t+12-m-(n+12)τ),
22.12.8 2Kdc(2Kt,k) =n=-πsin(π(t+12-nτ))
=n=-(m=-(-1)mt+12-m-nτ),
22.12.9 2Kknc(2Kt,k) =n=-(-1)nπsin(π(t+12-nτ))
=n=-(m=-(-1)m+nt+12-m-nτ),
22.12.10 -2Kksc(2Kt,k) =limNn=-NN(-1)nπtan(π(t+12-nτ))
=limNn=-NN(-1)n(limMm=-MM1t+12-m-nτ),
22.12.11 2Kns(2Kt,k) =n=-πsin(π(t-nτ))
=n=-(m=-(-1)mt-m-nτ),
22.12.12 2Kds(2Kt,k) =n=-(-1)nπsin(π(t-nτ))
=n=-(m=-(-1)m+nt-m-nτ),
22.12.13 2Kcs(2Kt,k) =limNn=-NN(-1)nπtan(π(t-nτ))
=limNn=-NN(-1)n(limMm=-MM1t-m-nτ).