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22 Jacobian Elliptic FunctionsProperties

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

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Keywords:
Eisenstein series, Jacobian elliptic functions, expansions in doubly-infinite partial fractions, trigonometric series expansions
Notes:
For the first right-hand sides of (22.12.2)–(22.12.4) see Lawden (1989, (8.8.3), (8.8.4), and (8.8.7)). The first right-hand sides of (22.12.5)–(22.12.13) can be obtained from the first right-hand sides of (22.12.2)–(22.12.4) by translations of the variable in accordance with the intersection of the first four rows and first four columns of Table 22.4.3. For example, in the case of (22.12.5) we apply the translation cd⁡(z,k)=sn⁡(z+K⁡,k) to (22.12.2). The second right-hand sides of (22.12.2)–(22.12.13) can be obtained from the corresponding first right-hand sides by substituting, as appropriate, the expansions π⁢csc⁡(π⁢ζ)=∑m=−∞∞(−1)m/(ζ−m) or π⁢cot⁡(π⁢ζ)=limM→∞∑m=−MM1/(ζ−m); compare (4.22.5) and (4.22.3). See also Walker (1996, §5.1) and Weil (1999, pp. 14–47).
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Annotations for Ch.22

With t∈ℂ and

22.12.1 τ=i⁢K′⁡(k)/K⁡(k),
22.12.2 2⁢K⁡⁢k⁢sn⁡(2⁢K⁡⁢t,k)=∑n=−∞∞πsin⁡(π⁢(t−(n+12)⁢τ))=∑n=−∞∞(∑m=−∞∞(−1)mt−m−(n+12)⁢τ),
22.12.3 2⁢i⁢K⁡⁢k⁢cn⁡(2⁢K⁡⁢t,k)=∑n=−∞∞(−1)n⁢πsin⁡(π⁢(t−(n+12)⁢τ))=∑n=−∞∞(∑m=−∞∞(−1)m+nt−m−(n+12)⁢τ),
22.12.4 2⁢i⁢K⁡⁢dn⁡(2⁢K⁡⁢t,k)=limN→∞∑n=−NN(−1)n⁢πtan⁡(π⁢(t−(n+12)⁢τ))=limN→∞∑n=−NN(−1)n⁢(limM→∞∑m=−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 2⁢K⁡⁢k⁢cd⁡(2⁢K⁡⁢t,k) =∑n=−∞∞πsin⁡(π⁢(t+12−(n+12)⁢τ))=∑n=−∞∞(∑m=−∞∞(−1)mt+12−m−(n+12)⁢τ),
22.12.6 −2⁢i⁢K⁡⁢k⁢k′⁢sd⁡(2⁢K⁡⁢t,k) =∑n=−∞∞(−1)n⁢πsin⁡(π⁢(t+12−(n+12)⁢τ))=∑n=−∞∞(∑m=−∞∞(−1)m+nt+12−m−(n+12)⁢τ),
22.12.7 2⁢i⁢K⁡⁢k′⁢nd⁡(2⁢K⁡⁢t,k) =limN→∞∑n=−NN(−1)n⁢πtan⁡(π⁢(t+12−(n+12)⁢τ))=limN→∞∑n=−NN(−1)n⁢limM→∞(∑m=−MM1t+12−m−(n+12)⁢τ),
22.12.8 2⁢K⁡⁢dc⁡(2⁢K⁡⁢t,k) =∑n=−∞∞πsin⁡(π⁢(t+12−n⁢τ))=∑n=−∞∞(∑m=−∞∞(−1)mt+12−m−n⁢τ),
22.12.9 2⁢K⁡⁢k′⁢nc⁡(2⁢K⁡⁢t,k) =∑n=−∞∞(−1)n⁢πsin⁡(π⁢(t+12−n⁢τ))=∑n=−∞∞(∑m=−∞∞(−1)m+nt+12−m−n⁢τ),
22.12.10 −2⁢K⁡⁢k′⁢sc⁡(2⁢K⁡⁢t,k) =limN→∞∑n=−NN(−1)n⁢πtan⁡(π⁢(t+12−n⁢τ))=limN→∞∑n=−NN(−1)n⁢(limM→∞∑m=−MM1t+12−m−n⁢τ),
22.12.11 2⁢K⁡⁢ns⁡(2⁢K⁡⁢t,k) =∑n=−∞∞πsin⁡(π⁢(t−n⁢τ))=∑n=−∞∞(∑m=−∞∞(−1)mt−m−n⁢τ),
22.12.12 2⁢K⁡⁢ds⁡(2⁢K⁡⁢t,k) =∑n=−∞∞(−1)n⁢πsin⁡(π⁢(t−n⁢τ))=∑n=−∞∞(∑m=−∞∞(−1)m+nt−m−n⁢τ),
22.12.13 2⁢K⁡⁢cs⁡(2⁢K⁡⁢t,k) =limN→∞∑n=−NN(−1)n⁢πtan⁡(π⁢(t−n⁢τ))=limN→∞∑n=−NN(−1)n⁢(limM→∞∑m=−MM1t−m−n⁢τ).
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