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Weierstrass zeta function

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11: 23.9 Laurent and Other Power Series
23.9.3 ζ ( z ) = 1 z - n = 2 c n 2 n - 1 z 2 n - 1 , 0 < | z | < | z 0 | .
12: 23.12 Asymptotic Approximations
23.12.2 ζ ( z ) = π 2 4 ω 1 2 ( z 3 + 2 ω 1 π cot ( π z 2 ω 1 ) - 8 ( z - ω 1 π sin ( π z ω 1 ) ) q 2 + O ( q 4 ) ) ,
13: 23.8 Trigonometric Series and Products
14: 19.25 Relations to Other Functions
19.25.37 ζ ( z + 2 ω ) + ( z + 2 ω ) ( z ) = ± 2 R G ( ( z ) - e 1 , ( z ) - e 2 , ( z ) - e 3 ) ,
19.25.39 ζ ( ω j ) + ω j e j = 2 R G ( 0 , e j - e k , e j - e ) ,
15: Errata
  • Equation (23.12.2)
    23.12.2 ζ ( z ) = π 2 4 ω 1 2 ( z 3 + 2 ω 1 π cot ( π z 2 ω 1 ) - 8 ( z - ω 1 π sin ( π z ω 1 ) ) q 2 + O ( q 4 ) )

    Originally, the factor of 2 was missing from the denominator of the argument of the cot function.

    Reported by Blagoje Oblak on 2019-05-27

  • Equation (19.25.37)

    The Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

  • 16: 23.21 Physical Applications
    §23.21 Physical Applications
    The Weierstrass function plays a similar role for cubic potentials in canonical form g 3 + g 2 x - 4 x 3 . …
    §23.21(ii) Nonlinear Evolution Equations
    §23.21(iii) Ellipsoidal Coordinates
  • String theory. See Green et al. (1988a, §8.2) and Polchinski (1998, §7.2).

  • 17: 22.1 Special Notation
    (For other notation see Notation for the Special Functions.) … The functions treated in this chapter are the three principal Jacobian elliptic functions sn ( z , k ) , cn ( z , k ) , dn ( z , k ) ; the nine subsidiary Jacobian elliptic functions cd ( z , k ) , sd ( z , k ) , nd ( z , k ) , dc ( z , k ) , nc ( z , k ) , sc ( z , k ) , ns ( z , k ) , ds ( z , k ) , cs ( z , k ) ; the amplitude function am ( x , k ) ; Jacobi’s epsilon and zeta functions ( x , k ) and Z ( x | k ) . … Similarly for the other functions.
    18: 23.23 Tables
    §23.23 Tables
    2 in Abramowitz and Stegun (1964) gives values of ( z ) , ( z ) , and ζ ( z ) to 7 or 8D in the rectangular and rhombic cases, normalized so that ω 1 = 1 and ω 3 = i a (rectangular case), or ω 1 = 1 and ω 3 = 1 2 + i a (rhombic case), for a = 1. …05, and in the case of ( z ) the user may deduce values for complex z by application of the addition theorem (23.10.1). Abramowitz and Stegun (1964) also includes other tables to assist the computation of the Weierstrass functions, for example, the generators as functions of the lattice invariants g 2 and g 3 . For earlier tables related to Weierstrass functions see Fletcher et al. (1962, pp. 503–505) and Lebedev and Fedorova (1960, pp. 223–226).
    19: 29.2 Differential Equations
    29.2.11 ζ = ( η ; g 2 , g 3 ) = ( η ) ,
    20: 31.2 Differential Equations
    Weierstrass’s Form
    ζ = i K + ξ ( e 1 - e 3 ) 1 / 2 ,
    where 2 ω 1 and 2 ω 3 with ( ω 3 / ω 1 ) > 0 are generators of the lattice 𝕃 for ( z | 𝕃 ) . …
    31.2.10 w ( ξ ) = ( ( ξ ) - e 3 ) ( 1 - 2 γ ) / 4 ( ( ξ ) - e 2 ) ( 1 - 2 δ ) / 4 ( ( ξ ) - e 1 ) ( 1 - 2 ϵ ) / 4 W ( ξ ) ,
    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 ,