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Weierstrass M-test

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11: 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
where x , y , z are the corresponding Cartesian coordinates and e 1 , e 2 , e 3 are constants. …
12: 23.6 Relations to Other Functions
§23.6(i) Theta Functions
§23.6(ii) Jacobian Elliptic Functions
§23.6(iii) General Elliptic Functions
§23.6(iv) Elliptic Integrals
13: 23.5 Special Lattices
§23.5(ii) Rectangular Lattice
In this case the lattice roots e 1 , e 2 , and e 3 are real and distinct. …
§23.5(iii) Lemniscatic Lattice
§23.5(iv) Rhombic Lattice
§23.5(v) Equianharmonic Lattice
14: 23.11 Integral Representations
§23.11 Integral Representations
23.11.2 ( z ) = 1 z 2 + 8 0 s ( e - s sinh 2 ( 1 2 z s ) f 1 ( s , τ ) + e i τ s sin 2 ( 1 2 z s ) f 2 ( s , τ ) ) d s ,
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 ,
15: 23.19 Interrelations
23.19.3 J ( τ ) = g 2 3 g 2 3 - 27 g 3 2 ,
where g 2 , g 3 are the invariants of the lattice 𝕃 with generators 1 and τ ; see §23.3(i). …
16: 23.12 Asymptotic Approximations
§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 ) ) ,
23.12.3 σ ( z ) = 2 ω 1 π exp ( π 2 z 2 24 ω 1 2 ) sin ( π z 2 ω 1 ) ( 1 - ( π 2 z 2 ω 1 2 - 4 sin 2 ( π z 2 ω 1 ) ) q 2 + O ( q 4 ) ) ,
17: 23.8 Trigonometric Series and Products
§23.8(i) Fourier Series
§23.8(ii) Series of Cosecants and Cotangents
where in (23.8.4) the terms in n and - n are to be bracketed together (the Eisenstein convention or principal value: see Weil (1999, p. 6) or Walker (1996, p. 3)). …
§23.8(iii) Infinite Products
18: 31.2 Differential Equations
Weierstrass’s Form
k 2 = ( e 2 - e 3 ) / ( e 1 - e 3 ) ,
e 1 = ( ω 1 ) ,
e 1 + e 2 + e 3 = 0 ,
where 2 ω 1 and 2 ω 3 with ( ω 3 / ω 1 ) > 0 are generators of the lattice 𝕃 for ( z | 𝕃 ) . …
19: 23.20 Mathematical Applications
§23.20 Mathematical Applications
§23.20(i) Conformal Mappings
§23.20(iii) Factorization
§23.20(v) Modular Functions and Number Theory
20: 23 Weierstrass Elliptic and Modular
Functions
Chapter 23 Weierstrass Elliptic and Modular Functions