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hyperelliptic functions

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1: 22.19 Physical Applications
§22.19(iv) Tops
The classical rotation of rigid bodies in free space or about a fixed point may be described in terms of elliptic, or hyperelliptic, functions if the motion is integrable (Audin (1999, Chapter 1)). Hyperelliptic functions u ( z ) are solutions of the equation z = 0 u ( f ( x ) ) - 1 / 2 d x , where f ( x ) is a polynomial of degree higher than 4. …
2: 21.7 Riemann Surfaces
§21.7(iii) Frobenius’ Identity
3: 19.16 Definitions
with the same conditions on x , y , z as for (19.16.1), but now z 0 . … and R D is a degenerate case of R J , so is R J a degenerate case of the hyperelliptic integral, …
§19.16(ii) R - a ( b ; z )
The R -function is often used to make a unified statement of a property of several elliptic integrals. …
4: 31.8 Solutions via Quadratures
ϵ = - m 3 + 1 2 , m 0 , m 1 , m 2 , m 3 = 0 , 1 , 2 , ,
31.8.2 w ± ( m ; λ ; z ) = Ψ g , N ( λ , z ) exp ( ± i ν ( λ ) 2 z 0 z t m 1 ( t - 1 ) m 2 ( t - a ) m 3 d t Ψ g , N ( λ , t ) t ( t - 1 ) ( t - a ) )
The variables λ and ν are two coordinates of the associated hyperelliptic (spectral) curve Γ : ν 2 = j = 1 2 g + 1 ( λ - λ j ) . … By automorphisms from §31.2(v), similar solutions also exist for m 0 , m 1 , m 2 , m 3 , and Ψ g , N ( λ , z ) may become a rational function in z . …For m = ( m 0 , 0 , 0 , 0 ) , these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …