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1: 23.8 Trigonometric Series and Products
23.8.1 ( z ) + η 1 ω 1 π 2 4 ω 1 2 csc 2 ( π z 2 ω 1 ) = 2 π 2 ω 1 2 n = 1 n q 2 n 1 q 2 n cos ( n π z ω 1 ) ,
23.8.5 η 1 = π 2 2 ω 1 ( 1 6 + n = 1 csc 2 ( n π ω 3 ω 1 ) ) ,
2: 1.9 Calculus of a Complex Variable
§1.9(i) Complex Numbers
Polar Representation
Modulus and Phase
Complex Conjugate
Powers
3: 23.2 Definitions and Periodic Properties
If ω 1 and ω 3 are nonzero real or complex numbers such that ( ω 3 / ω 1 ) > 0 , then the set of points 2 m ω 1 + 2 n ω 3 , with m , n , constitutes a lattice 𝕃 with 2 ω 1 and 2 ω 3 lattice generators. … where …
23.2.13 η 1 + η 2 + η 3 = 0 ,
4: 20.12 Mathematical Applications
The space of complex tori / ( + τ ) (that is, the set of complex numbers z in which two of these numbers z 1 and z 2 are regarded as equivalent if there exist integers m , n such that z 1 z 2 = m + τ n ) is mapped into the projective space P 3 via the identification z ( θ 1 ( 2 z | τ ) , θ 2 ( 2 z | τ ) , θ 3 ( 2 z | τ ) , θ 4 ( 2 z | τ ) ) . …
5: 1.1 Special Notation
In the physics, applied maths, and engineering literature a common alternative to a ¯ is a , a being a complex number or a matrix; the Hermitian conjugate of 𝐀 is usually being denoted 𝐀 .
6: 23.12 Asymptotic Approximations
Also,
23.12.4 η 1 = π 2 4 ω 1 ( 1 3 8 q 2 + O ( q 4 ) ) ,
7: 23.5 Special Lattices
The parallelogram 0 , 2 ω 1 , 2 ω 1 + 2 ω 3 , 2 ω 3 is a square, and
8: 19.25 Relations to Other Functions
19.25.38 ω j = R F ( 0 , e j e k , e j e ) ,
19.25.39 ζ ( ω j ) + ω j e j = 2 R G ( 0 , e j e k , e j e ) ,
19.25.41 σ j ( z ) = exp ( η j z ) σ ( z + ω j ) / σ ( ω j ) , j = 1 , 2 , 3 ,
9: 23.10 Addition Theorems and Other Identities
23.10.12 n ζ ( n z ) = n ( n 1 ) ( η 1 + η 3 ) + j = 0 n 1 = 0 n 1 ζ ( z + 2 j n ω 1 + 2 n ω 3 ) ,
23.10.13 σ ( n z ) = A n e n ( n 1 ) ( η 1 + η 3 ) z j = 0 n 1 = 0 n 1 σ ( z + 2 j n ω 1 + 2 n ω 3 ) ,
23.10.15 A n = ( π 2 G 2 ω 1 ) n 2 1 q n ( n 1 ) / 2 i n 1 exp ( ( n 1 ) η 1 3 ω 1 ( ( 2 n 1 ) ( ω 1 2 + ω 3 2 ) + 3 ( n 1 ) ω 1 ω 3 ) ) ,
10: 31.2 Differential Equations
31.2.11 d 2 W / d ξ 2 + ( H + b 0 ( ξ ) + b 1 ( ξ + ω 1 ) + b 2 ( ξ + ω 2 ) + b 3 ( ξ + ω 3 ) ) W = 0 ,