hyperelliptic functions

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1: 22.19 Physical Applications

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§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 $z={\int }_0^u{(f(x))}^{-1/2}dx$, where $f(x)$ is a polynomial of degree higher than 4. …

2: 21.7 Riemann Surfaces

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§21.7(iii) Frobenius’ Identity

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3: 19.16 Definitions

… ►with the same conditions on $x$, $y$, $z$ as for (19.16.1), but now $z\ne 0$. … ►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

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$ϵ=-m_3+\frac{1}{2}$ , $m_0,m_1,m_2,m_3=0,1,2,\mathrm{\dots }$,

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31.8.2 $$w_{\pm }(𝐦;\lambda ;z)=\sqrt{{\mathrm{\Psi }}_{g,N}(\lambda ,z)}\exp \left(\pm \frac{\mathrm{i}\nu (\lambda )}{2}{\int }_{z_0}^z\frac{t^{m_1}{(t-1)}^{m_2}{(t-a)}^{m_3}dt}{{\mathrm{\Psi }}_{g,N}(\lambda ,t)\sqrt{t(t-1)(t-a)}}\right)$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\nu$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\lambda =-4q$ , ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$: polynomial, $g$: degree of $\lambda$ and $N$: degree of $z$

Referenced by:

§31.8

Permalink:

http://dlmf.nist.gov/31.8.E2

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See also:

Annotations for §31.8 and Ch.31

… ►The variables $\lambda$ and $\nu$ are two coordinates of the associated hyperelliptic (spectral) curve $\mathrm{\Gamma }:{\nu }^2={\prod }_{j=1}^{2g+1}(\lambda -{\lambda }_j)$. … ►By automorphisms from §31.2(v), similar solutions also exist for $m_0,m_1,m_2,m_3\in \mathbb{Z}$, and ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ 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. …