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Weierstrass M-test

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21: 19.25 Relations to Other Functions
§19.25(vi) Weierstrass Elliptic Functions
Let 𝕃 be a lattice for the Weierstrass elliptic function ( z ) . …The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which ( z ) - e j < 0 , for some j . … for some 2 ω j 𝕃 and ( ω j ) = e j . … in which 2 ω 1 and 2 ω 3 are generators for the lattice 𝕃 , ω 2 = - ω 1 - ω 3 , and η j = ζ ( ω j ) (see (23.2.12)). …
22: 29.2 Differential Equations
we have
29.2.9 d 2 w d η 2 + ( g - ν ( ν + 1 ) ( η ) ) w = 0 ,
29.2.11 ζ = ( η ; g 2 , g 3 ) = ( η ) ,
For the Weierstrass function see §23.2(ii). …
23: 23.22 Methods of Computation
§23.22 Methods of Computation
§23.22(ii) Lattice Calculations
The corresponding values of e 1 , e 2 , e 3 are calculated from (23.6.2)–(23.6.4), then g 2 and g 3 are obtained from (23.3.6) and (23.3.7). … Suppose that the invariants g 2 = c , g 3 = d , are given, for example in the differential equation (23.3.10) or via coefficients of an elliptic curve (§23.20(ii)). … Assume c = g 2 = - 4 ( 3 - 2 i ) and d = g 3 = 4 ( 4 - 2 i ) . …
24: Peter L. Walker
25: 19.10 Relations to Other Functions
§19.10(i) Theta and Elliptic Functions
For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …
26: 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 ) .
27: William P. Reinhardt
28: 20.9 Relations to Other Functions
§20.9(ii) Elliptic Functions and Modular Functions
See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. …
29: Bibliography E
  • U. Eckhardt (1980) Algorithm 549: Weierstrass’ elliptic functions. ACM Trans. Math. Software 6 (1), pp. 112–120.
  • M. Eichler and D. Zagier (1982) On the zeros of the Weierstrass -function. Math. Ann. 258 (4), pp. 399–407.
  • 30: 22.1 Special Notation