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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)mtm(n+12)τ),
22.12.3 2iKkcn(2Kt,k)=n=(1)nπsin(π(t(n+12)τ))=n=(m=(1)m+ntm(n+12)τ),
22.12.4 2iKdn(2Kt,k)=limNn=NN(1)nπtan(π(t(n+12)τ))=limNn=NN(1)n(limMm=MM1tm(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+12m(n+12)τ),
22.12.6 2iKkksd(2Kt,k) =n=(1)nπsin(π(t+12(n+12)τ))=n=(m=(1)m+nt+12m(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+12m(n+12)τ),
22.12.8 2Kdc(2Kt,k) =n=πsin(π(t+12nτ))=n=(m=(1)mt+12mnτ),
22.12.9 2Kknc(2Kt,k) =n=(1)nπsin(π(t+12nτ))=n=(m=(1)m+nt+12mnτ),
22.12.10 2Kksc(2Kt,k) =limNn=NN(1)nπtan(π(t+12nτ))=limNn=NN(1)n(limMm=MM1t+12mnτ),
22.12.11 2Kns(2Kt,k) =n=πsin(π(tnτ))=n=(m=(1)mtmnτ),
22.12.12 2Kds(2Kt,k) =n=(1)nπsin(π(tnτ))=n=(m=(1)m+ntmnτ),
22.12.13 2Kcs(2Kt,k) =limNn=NN(1)nπtan(π(tnτ))=limNn=NN(1)n(limMm=MM1tmnτ).