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1: 17.14 Constant Term Identities
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17.14.2 n = 0 q n ⁒ ( n + 1 ) ( q 2 ; q 2 ) n ⁒ ( q ; q 2 ) n + 1 =  coeff. of  ⁒ z 0 ⁒  in  ⁒ ( z ⁒ q ; q 2 ) ⁒ ( z 1 ⁒ q ; q 2 ) ⁒ ( q 2 ; q 2 ) ( z 1 ⁒ q 2 ; q 2 ) ⁒ ( q ; q 2 ) ⁒ ( z 1 ⁒ q ; q 2 ) = 1 ( q ; q 2 ) ⁒  coeff. of  ⁒ z 0 ⁒  in  ⁒ ( z ⁒ q ; q 2 ) ⁒ ( z 1 ⁒ q ; q 2 ) ⁒ ( q 2 ; q 2 ) ( z 1 ⁒ q ; q ) = H ⁑ ( q ) ( q ; q 2 ) ,
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17.14.3 n = 0 q n ⁒ ( n + 1 ) ( q 2 ; q 2 ) n ⁒ ( q ; q 2 ) n + 1 =  coeff. of  ⁒ z 0 ⁒  in  ⁒ ( z ⁒ q ; q 2 ) ⁒ ( z 1 ⁒ q ; q 2 ) ⁒ ( q 2 ; q 2 ) ( z 1 ; q 2 ) ⁒ ( q ; q 2 ) ⁒ ( z 1 ⁒ q ; q 2 ) = 1 ( q ; q 2 ) ⁒  coeff. of  ⁒ z 0 ⁒  in  ⁒ ( z ⁒ q ; q 2 ) ⁒ ( z 1 ⁒ q ; q 2 ) ⁒ ( q 2 ; q 2 ) ( z 1 ; q ) = G ⁑ ( q ) ( q ; q 2 ) ,
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17.14.4 n = 0 q n 2 ( q 2 ; q 2 ) n ⁒ ( q ; q 2 ) n =  coeff. of  ⁒ z 0 ⁒  in  ⁒ ( z ⁒ q ; q 2 ) ⁒ ( z 1 ⁒ q ; q 2 ) ⁒ ( q 2 ; q 2 ) ( z 1 ; q 2 ) ⁒ ( q ; q 2 ) ⁒ ( z 1 ; q 2 ) = 1 ( q ; q 2 ) ⁒  coeff. of  ⁒ z 0 ⁒  in  ⁒ ( z ⁒ q ; q 2 ) ⁒ ( z 1 ⁒ q ; q 2 ) ⁒ ( q 2 ; q 2 ) ( z 2 ; q 4 ) = G ⁑ ( q 4 ) ( q ; q 2 ) ,
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17.14.5 n = 0 q n 2 + 2 ⁒ n ( q 2 ; q 2 ) n ⁒ ( q ; q 2 ) n + 1 =  coeff. of  ⁒ z 0 ⁒  in  ⁒ ( z ⁒ q ; q 2 ) ⁒ ( z 1 ⁒ q ; q 2 ) ⁒ ( q 2 ; q 2 ) ( q 2 ⁒ z 1 ; q 2 ) ⁒ ( q ; q 2 ) ⁒ ( z 1 ⁒ q 2 ; q 2 ) = 1 ( q ; q 2 ) ⁒  coeff. of  ⁒ z 0 ⁒  in  ⁒ ( z ⁒ q ; q 2 ) ⁒ ( z 1 ⁒ q ; q 2 ) ⁒ ( q 2 ; q 2 ) ( q 4 ⁒ z 2 ; q 4 ) = H ⁑ ( q 4 ) ( q ; q 2 ) .
2: 4.31 Special Values and Limits
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Table 4.31.1: Hyperbolic functions: values at multiples of 1 2 ⁒ Ο€ ⁒ i .
β–Ί β–Ίβ–Ίβ–Ίβ–Ίβ–Ίβ–Ίβ–Ί
z 0 1 2 ⁒ Ο€ ⁒ i Ο€ ⁒ i 3 2 ⁒ Ο€ ⁒ i
cosh ⁑ z 1 0 1 0
tanh ⁑ z 0 ⁒ i 0 ⁒ i 1
csch ⁑ z i i 0
sech ⁑ z 1 1 0
coth ⁑ z 0 0 1
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3: 17.13 Integrals
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17.13.1 c d ( q ⁒ x / c ; q ) ⁒ ( q ⁒ x / d ; q ) ( a ⁒ x / c ; q ) ⁒ ( b ⁒ x / d ; q ) ⁒ d q x = ( 1 q ) ⁒ ( q ; q ) ⁒ ( a ⁒ b ; q ) ⁒ c ⁒ d ⁒ ( c / d ; q ) ⁒ ( d / c ; q ) ( a ; q ) ⁒ ( b ; q ) ⁒ ( c + d ) ⁒ ( b ⁒ c / d ; q ) ⁒ ( a ⁒ d / c ; q ) ,
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17.13.2 c d ( q ⁒ x / c ; q ) ⁒ ( q ⁒ x / d ; q ) ( x ⁒ q Ξ± / c ; q ) ⁒ ( x ⁒ q Ξ² / d ; q ) ⁒ d q x = Ξ“ q ⁑ ( Ξ± ) ⁒ Ξ“ q ⁑ ( Ξ² ) Ξ“ q ⁑ ( Ξ± + Ξ² ) ⁒ c ⁒ d c + d ⁒ ( c / d ; q ) ⁒ ( d / c ; q ) ( q Ξ² ⁒ c / d ; q ) ⁒ ( q Ξ± ⁒ d / c ; q ) .
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17.13.3 0 t Ξ± 1 ⁒ ( t ⁒ q Ξ± + Ξ² ; q ) ( t ; q ) ⁒ d t = Ξ“ ⁑ ( Ξ± ) ⁒ Ξ“ ⁑ ( 1 Ξ± ) ⁒ Ξ“ q ⁑ ( Ξ² ) Ξ“ q ⁑ ( 1 Ξ± ) ⁒ Ξ“ q ⁑ ( Ξ± + Ξ² ) ,
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17.13.4 0 t Ξ± 1 ⁒ ( c ⁒ t ⁒ q Ξ± + Ξ² ; q ) ( c ⁒ t ; q ) ⁒ d q t = Ξ“ q ⁑ ( Ξ± ) ⁒ Ξ“ q ⁑ ( Ξ² ) ⁒ ( c ⁒ q Ξ± ; q ) ⁒ ( q 1 Ξ± / c ; q ) Ξ“ q ⁑ ( Ξ± + Ξ² ) ⁒ ( c ; q ) ⁒ ( q / c ; q ) .
4: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
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22.12.2 2 ⁒ K ⁑ ⁒ k ⁒ sn ⁑ ( 2 ⁒ K ⁑ ⁒ t , k ) = n = Ο€ sin ⁑ ( Ο€ ⁒ ( t ( n + 1 2 ) ⁒ Ο„ ) ) = n = ( m = ( 1 ) m t m ( n + 1 2 ) ⁒ Ο„ ) ,
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22.12.3 2 ⁒ i ⁒ K ⁑ ⁒ k ⁒ cn ⁑ ( 2 ⁒ K ⁑ ⁒ t , k ) = n = ( 1 ) n ⁒ Ο€ sin ⁑ ( Ο€ ⁒ ( t ( n + 1 2 ) ⁒ Ο„ ) ) = n = ( m = ( 1 ) m + n t m ( n + 1 2 ) ⁒ Ο„ ) ,
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22.12.8 2 ⁒ K ⁑ ⁒ dc ⁑ ( 2 ⁒ K ⁑ ⁒ t , k ) = n = Ο€ sin ⁑ ( Ο€ ⁒ ( t + 1 2 n ⁒ Ο„ ) ) = n = ( m = ( 1 ) m t + 1 2 m n ⁒ Ο„ ) ,
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22.12.11 2 ⁒ K ⁑ ⁒ ns ⁑ ( 2 ⁒ K ⁑ ⁒ t , k ) = n = Ο€ sin ⁑ ( Ο€ ⁒ ( t n ⁒ Ο„ ) ) = n = ( m = ( 1 ) m t m n ⁒ Ο„ ) ,
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22.12.12 2 ⁒ K ⁑ ⁒ ds ⁑ ( 2 ⁒ K ⁑ ⁒ t , k ) = n = ( 1 ) n ⁒ Ο€ sin ⁑ ( Ο€ ⁒ ( t n ⁒ Ο„ ) ) = n = ( m = ( 1 ) m + n t m n ⁒ Ο„ ) ,
5: 24.20 Tables
β–ΊAbramowitz and Stegun (1964, Chapter 23) includes exact values of k = 1 m k n , m = 1 ⁒ ( 1 ) ⁒ 100 , n = 1 ⁒ ( 1 ) ⁒ 10 ; k = 1 k n , k = 1 ( 1 ) k 1 ⁒ k n , k = 0 ( 2 ⁒ k + 1 ) n , n = 1 , 2 , , 20D; k = 0 ( 1 ) k ⁒ ( 2 ⁒ k + 1 ) n , n = 1 , 2 , , 18D. …
6: 4.22 Infinite Products and Partial Fractions
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4.22.1 sin ⁑ z = z ⁒ n = 1 ( 1 z 2 n 2 ⁒ Ο€ 2 ) ,
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4.22.2 cos ⁑ z = n = 1 ( 1 4 ⁒ z 2 ( 2 ⁒ n 1 ) 2 ⁒ Ο€ 2 ) .
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4.22.3 cot ⁑ z = 1 z + 2 ⁒ z ⁒ n = 1 1 z 2 n 2 ⁒ Ο€ 2 ,
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4.22.4 csc 2 ⁑ z = n = 1 ( z n ⁒ Ο€ ) 2 ,
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4.22.5 csc ⁑ z = 1 z + 2 ⁒ z ⁒ n = 1 ( 1 ) n z 2 n 2 ⁒ Ο€ 2 .
7: 4.36 Infinite Products and Partial Fractions
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4.36.1 sinh ⁑ z = z ⁒ n = 1 ( 1 + z 2 n 2 ⁒ Ο€ 2 ) ,
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4.36.2 cosh ⁑ z = n = 1 ( 1 + 4 ⁒ z 2 ( 2 ⁒ n 1 ) 2 ⁒ Ο€ 2 ) .
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4.36.3 coth ⁑ z = 1 z + 2 ⁒ z ⁒ n = 1 1 z 2 + n 2 ⁒ Ο€ 2 ,
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4.36.4 csch 2 ⁑ z = n = 1 ( z n ⁒ Ο€ ⁒ i ) 2 ,
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4.36.5 csch ⁑ z = 1 z + 2 ⁒ z ⁒ n = 1 ( 1 ) n z 2 + n 2 ⁒ Ο€ 2 .
8: 27.7 Lambert Series as Generating Functions
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27.7.1 n = 1 f ⁑ ( n ) ⁒ x n 1 x n .
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27.7.2 n = 1 f ⁑ ( n ) ⁒ x n 1 x n = n = 1 d | n f ⁑ ( d ) ⁒ x n .
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27.7.3 n = 1 μ ⁑ ( n ) ⁒ x n 1 x n = x ,
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27.7.5 n = 1 n Ξ± ⁒ x n 1 x n = n = 1 Οƒ Ξ± ⁑ ( n ) ⁒ x n ,
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27.7.6 n = 1 λ ⁑ ( n ) ⁒ x n 1 x n = n = 1 x n 2 .
9: 32.6 Hamiltonian Structure
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32.6.16 z ⁒ H III ⁑ ( q , p , z ) = q 2 ⁒ p 2 ( κ ⁒ z ⁒ q 2 + ( 2 ⁒ θ 0 + 1 ) ⁒ q κ 0 ⁒ z ) ⁒ p + κ ⁒ ( θ 0 + θ ) ⁒ z ⁒ q ,
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32.6.18 z ⁒ p = 2 ⁒ q ⁒ p 2 + 2 ⁒ κ ⁒ z ⁒ q ⁒ p + ( 2 ⁒ θ 0 + 1 ) ⁒ p κ ⁒ ( θ 0 + θ ) ⁒ z .
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32.6.19 ( α , β , γ , δ ) = ( 2 ⁒ κ ⁒ θ , 2 ⁒ κ 0 ⁒ ( θ 0 + 1 ) , κ 2 , κ 0 2 ) .
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32.6.21 ( z ⁒ Οƒ ′′ Οƒ ) 2 + 2 ⁒ ( ( Οƒ ) 2 ΞΊ 0 2 ⁒ ΞΊ 2 ⁒ z 2 ) ⁒ ( z ⁒ Οƒ 2 ⁒ Οƒ ) + 8 ⁒ ΞΊ 0 ⁒ ΞΊ ⁒ ΞΈ 0 ⁒ ΞΈ ⁒ z ⁒ Οƒ = 4 ⁒ ΞΊ 0 2 ⁒ ΞΊ 2 ⁒ ( ΞΈ 0 2 + ΞΈ 2 ) ⁒ z 2 .
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32.6.29 ΞΆ 2 ⁒ ( Οƒ ′′ ) 2 + ( 4 ⁒ ( Οƒ ) 2 Ξ· 0 2 ⁒ Ξ· 2 ) ⁒ ( ΞΆ ⁒ Οƒ Οƒ ) + Ξ· 0 ⁒ Ξ· ⁒ ΞΈ 0 ⁒ ΞΈ ⁒ Οƒ = 1 4 ⁒ Ξ· 0 2 ⁒ Ξ· 2 ⁒ ( ΞΈ 0 2 + ΞΈ 2 ) .
10: 20.6 Power Series
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20.6.2 ΞΈ 1 ⁑ ( Ο€ ⁒ z | Ο„ ) = Ο€ ⁒ z ⁒ ΞΈ 1 ⁑ ( 0 | Ο„ ) ⁒ exp ⁑ ( j = 1 1 2 ⁒ j ⁒ Ξ΄ 2 ⁒ j ⁑ ( Ο„ ) ⁒ z 2 ⁒ j ) ,
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20.6.6 Ξ΄ 2 ⁒ j ⁑ ( Ο„ ) = n = m = | m | + | n | 0 ( m + n ⁒ Ο„ ) 2 ⁒ j ,
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20.6.7 Ξ± 2 ⁒ j ⁑ ( Ο„ ) = n = m = ( m 1 2 + n ⁒ Ο„ ) 2 ⁒ j ,
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20.6.8 Ξ² 2 ⁒ j ⁑ ( Ο„ ) = n = m = ( m 1 2 + ( n 1 2 ) ⁒ Ο„ ) 2 ⁒ j ,
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20.6.9 Ξ³ 2 ⁒ j ⁑ ( Ο„ ) = n = m = ( m + ( n 1 2 ) ⁒ Ο„ ) 2 ⁒ j ,