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1: 32.6 Hamiltonian Structure
The function
32.6.7 q = - σ ,
32.6.8 p = - σ ′′ ,
The function σ ( z ) = H II ( q , p , z ) defined by (32.6.9) satisfies … The function
2: 23.1 Special Notation
The main functions treated in this chapter are the Weierstrass -function ( z ) = ( z | 𝕃 ) = ( z ; g 2 , g 3 ) ; the Weierstrass zeta function ζ ( z ) = ζ ( z | 𝕃 ) = ζ ( z ; g 2 , g 3 ) ; the Weierstrass sigma function σ ( z ) = σ ( z | 𝕃 ) = σ ( z ; g 2 , g 3 ) ; the elliptic modular function λ ( τ ) ; Klein’s complete invariant J ( τ ) ; Dedekind’s eta function η ( τ ) . …
3: 23.2 Definitions and Periodic Properties
§23.2(ii) Weierstrass Elliptic Functions
23.2.6 σ ( z ) = z w 𝕃 { 0 } ( ( 1 - z w ) exp ( z w + z 2 2 w 2 ) ) .
The function σ ( z ) is entire and odd, with simple zeros at the lattice points. … For j = 1 , 2 , 3 , the function σ ( z ) satisfies … For further quasi-periodic properties of the σ -function see Lawden (1989, §6.2).
4: 10.73 Physical Applications
For applications of the Rayleigh function σ n ( ν ) 10.21(xiii)) to problems of heat conduction and diffusion in liquids see Kapitsa (1951a).
5: 23.10 Addition Theorems and Other Identities
23.10.3 σ ( u + v ) σ ( u - v ) σ 2 ( u ) σ 2 ( v ) = ( v ) - ( u ) ,
23.10.4 σ ( u + v ) σ ( u - v ) σ ( x + y ) σ ( x - y ) + σ ( v + x ) σ ( v - x ) σ ( u + y ) σ ( u - y ) + σ ( x + u ) σ ( x - u ) σ ( v + y ) σ ( v - y ) = 0 .
For further addition-type identities for the σ -function see Lawden (1989, §6.4). …
23.10.19 σ ( c z | c 𝕃 ) = c σ ( z | 𝕃 ) .
6: 33.13 Complex Variable and Parameters
7: 26.2 Basic Definitions
The function σ also interchanges 3 and 6, and sends 4 to itself. …
8: 10.21 Zeros
If σ ν is a zero of 𝒞 ν ( z ) , then … The parameter t may be regarded as a continuous variable and ρ ν , σ ν as functions ρ ν ( t ) , σ ν ( t ) of t . … The functions ρ ν ( t ) and σ ν ( t ) are related to the inverses of the phase functions θ ν ( x ) and ϕ ν ( x ) defined in §10.18(i): if ν 0 , then … The Rayleigh function σ n ( ν ) is defined by
10.21.55 σ n ( ν ) = m = 1 ( j ν , m ) - 2 n , n = 1 , 2 , 3 , .
9: 23.4 Graphics
Line graphs of the Weierstrass functions ( x ) , ζ ( x ) , and σ ( x ) , illustrating the lemniscatic and equianharmonic cases. … Surfaces for the Weierstrass functions ( z ) , ζ ( z ) , and σ ( z ) . …
10: 23.6 Relations to Other Functions
For further results for the σ -function see Lawden (1989, §6.2). … For representations of the Jacobi functions sn , cn , and dn as quotients of σ -functions see Lawden (1989, §§6.2, 6.3). … For representations of general elliptic functions23.2(iii)) in terms of σ ( z ) and ( z ) see Lawden (1989, §§8.9, 8.10), and for expansions in terms of ζ ( z ) see Lawden (1989, §8.11). …