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Weierstrass M-test

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1: 23.2 Definitions and Periodic Properties
§23.2(i) Lattices
§23.2(ii) Weierstrass Elliptic Functions
§23.2(iii) Periodicity
2: 23.3 Differential Equations
The lattice invariants are defined by … and are denoted by e 1 , e 2 , e 3 . … Similarly for ζ ( z ; g 2 , g 3 ) and σ ( z ; g 2 , g 3 ) . 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. …
§23.3(ii) Differential Equations and Derivatives
3: 23.14 Integrals
§23.14 Integrals
23.14.2 2 ( z ) d z = 1 6 ( z ) + 1 12 g 2 z ,
4: 23.13 Zeros
§23.13 Zeros
For information on the zeros of ( z ) see Eichler and Zagier (1982).
5: 23.10 Addition Theorems and Other Identities
§23.10(i) Addition Theorems
§23.10(ii) Duplication Formulas
(23.10.8) continues to hold when e 1 , e 2 , e 3 are permuted cyclically. …
§23.10(iii) n -Tuple Formulas
§23.10(iv) Homogeneity
6: 23.4 Graphics
§23.4(i) Real Variables
Line graphs of the Weierstrass functions ( x ) , ζ ( x ) , and σ ( x ) , illustrating the lemniscatic and equianharmonic cases. …
See accompanying text
Figure 23.4.6: σ ( x ; 0 , g 3 ) for - 5 x 5 , g 3 = 0. … Magnify
§23.4(ii) Complex Variables
Surfaces for the Weierstrass functions ( z ) , ζ ( z ) , and σ ( z ) . …
7: 23.7 Quarter Periods
§23.7 Quarter Periods
23.7.2 ( 1 2 ω 2 ) = e 2 - i ( e 1 - e 2 ) ( e 2 - e 3 ) = e 2 - i ω 1 - 2 ( K ( k ) ) 2 k k ,
23.7.3 ( 1 2 ω 3 ) = e 3 - ( e 1 - e 3 ) ( e 2 - e 3 ) = e 3 - ω 1 - 2 ( K ( k ) ) 2 k ,
8: 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).
9: 23.9 Laurent and Other Power Series
§23.9 Laurent and Other Power Series
c 2 = 1 20 g 2 ,
For j = 1 , 2 , 3 , and with e j as in §23.3(i),
23.9.6 ( ω j + t ) = e j + ( 3 e j 2 - 5 c 2 ) t 2 + ( 10 c 2 e j + 21 c 3 ) t 4 + ( 7 c 2 e j 2 + 21 c 3 e j + 5 c 2 2 ) t 6 + O ( t 8 ) ,
Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as 1 / ( z ) 0 . …
10: 23.1 Special Notation
𝕃 lattice in .
= e i π τ nome.
Δ discriminant g 2 3 - 27 g 3 2 .
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 ) . …