# 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 $z=\int_{0}^{u}(f(x))^{-1/2}\,\mathrm{d}x$, where $f(x)$ is a polynomial of degree higher than 4. …
##### 3: 19.16 Definitions
with the same conditions on $x$, $y$, $z$ as for (19.16.1), but now $z\neq 0$. … and $R_{D}$ is a degenerate case of $R_{J}$, so is $R_{J}$ a degenerate case of the hyperelliptic integral, …
###### §19.16(ii) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$
The $R$-function is often used to make a unified statement of a property of several elliptic integrals. …
##### 4: 31.8 Solutions via Quadratures
$\epsilon=-m_{3}+\tfrac{1}{2}$ , $m_{0},m_{1},m_{2},m_{3}=0,1,2,\dots$,
31.8.2 $w_{\pm}(\mathbf{m};\lambda;z)=\sqrt{\Psi_{g,N}(\lambda,z)}\*\exp\left(\pm\frac% {i\nu(\lambda)}{2}\int_{z_{0}}^{z}\frac{t^{m_{1}}(t-1)^{m_{2}}(t-a)^{m_{3}}\,% \mathrm{d}t}{\Psi_{g,N}(\lambda,t)\sqrt{t(t-1)(t-a)}}\right)$
The variables $\lambda$ and $\nu$ are two coordinates of the associated hyperelliptic (spectral) curve $\Gamma:\nu^{2}=\prod_{j=1}^{2g+1}(\lambda-\lambda_{j})$. … By automorphisms from §31.2(v), similar solutions also exist for $m_{0},m_{1},m_{2},m_{3}\in\mathbb{Z}$, and $\Psi_{g,N}(\lambda,z)$ may become a rational function in $z$. …For $\mathbf{m}=(m_{0},0,0,0)$, these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …