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Weierstrass zeta function

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1: 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 ) . …
2: 23.14 Integrals
3: 23.2 Definitions and Periodic Properties
23.2.5 ζ ( z ) = 1 z + w 𝕃 { 0 } ( 1 z - w + 1 w + z w 2 ) ,
( z ) and ζ ( z ) are meromorphic functions with poles at the lattice points. … The function ζ ( z ) is quasi-periodic: for j = 1 , 2 , 3 , …
4: 25.1 Special Notation
The main related functions are the Hurwitz zeta function ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
5: 23.3 Differential Equations
As functions of g 2 and g 3 , ( z ; g 2 , g 3 ) and ζ ( z ; g 2 , g 3 ) are meromorphic and σ ( z ; g 2 , g 3 ) is entire. …
6: 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 ) . …
7: 23.11 Integral Representations
23.11.3 ζ ( z ) = 1 z + 0 ( e - s ( z s - sinh ( z s ) ) f 1 ( s , τ ) - e i τ s ( z s - sin ( z s ) ) f 2 ( s , τ ) ) d s ,
8: 23.6 Relations to Other Functions
23.6.27 ζ ( z | 𝕃 1 ) - ζ ( z + 2 K | 𝕃 1 ) + ζ ( 2 K | 𝕃 1 ) = ns ( z , k ) ,
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). …
9: 23.10 Addition Theorems and Other Identities
23.10.2 ζ ( u + v ) = ζ ( u ) + ζ ( v ) + 1 2 ζ ′′ ( u ) - ζ ′′ ( v ) ζ ( u ) - ζ ( v ) ,
23.10.6 ( ζ ( u ) + ζ ( v ) + ζ ( w ) ) 2 + ζ ( u ) + ζ ( v ) + ζ ( w ) = 0 .
23.10.9 ζ ( 2 z ) = 2 ζ ( z ) + 1 2 ζ ′′′ ( z ) ζ ′′ ( z ) ,
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 ) ,
23.10.18 ζ ( c z | c 𝕃 ) = c - 1 ζ ( z | 𝕃 ) ,
10: 23.22 Methods of Computation
The functions ζ ( z ) and σ ( z ) are computed in a similar manner: the former by replacing u and z in (23.6.13) by z and π z / ( 2 ω 1 ) , respectively, and also referring to (23.6.8); the latter by applying (23.6.9). …