hyperelliptic integrals

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… ►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, …

… ►

… ►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. …

… ►

21.7.6

\Omega_{jk}=\oint_{b_{k}}\omega_{j},

j,k=1,2,\dots,g

ⓘ

Symbols:: $g$ : positive integer, $\omega$ : differential and $b_{j}$ : cycles
Permalink:: http://dlmf.nist.gov/21.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(i), §21.7 and Ch.21

… ►

21.7.9

E(P_{1},P_{2})=\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{% \boldsymbol{{\beta}}}\left(\int_{P_{1}}^{P_{2}}\boldsymbol{{\omega}}\middle|% \boldsymbol{{\Omega}}\right)\Bigg{/}\left(\sqrt{\zeta(P_{1})}\sqrt{\zeta(P_{2}% )}\right),

ⓘ

Defines:: $E(\NVar{P_{1}},\NVar{P_{2}})$ : prime form (locally)
Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\int$ : integral, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector, $P_{1},P_{2}$ : points and $\boldsymbol{{\omega}}$ : vector of differentials
Permalink:: http://dlmf.nist.gov/21.7.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(ii), §21.7 and Ch.21

… ►

21.7.10

\theta\left(\mathbf{z}+\int_{P_{1}}^{P_{3}}\boldsymbol{{\omega}}\middle|% \boldsymbol{{\Omega}}\right)\theta\left(\mathbf{z}+\int_{P_{2}}^{P_{4}}% \boldsymbol{{\omega}}\middle|\boldsymbol{{\Omega}}\right)E(P_{3},P_{2})E(P_{1}% ,P_{4})+\theta\left(\mathbf{z}+\int_{P_{2}}^{P_{3}}\boldsymbol{{\omega}}% \middle|\boldsymbol{{\Omega}}\right)\theta\left(\mathbf{z}+\int_{P_{1}}^{P_{4}% }\boldsymbol{{\omega}}\middle|\boldsymbol{{\Omega}}\right)E(P_{3},P_{1})E(P_{4% },P_{2})=\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)\theta\left% (\mathbf{z}+\int_{P_{1}}^{P_{3}}\boldsymbol{{\omega}}+\int_{P_{2}}^{P_{4}}% \boldsymbol{{\omega}}\middle|\boldsymbol{{\Omega}}\right)E(P_{1},P_{2})E(P_{3}% ,P_{4}),

ⓘ

Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\int$ : integral, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $E(\NVar{P_{1}},\NVar{P_{2}})$ : prime form, $P_{1},P_{2}$ : points and $\boldsymbol{{\omega}}$ : vector of differentials
Referenced by:: §21.7(ii)
Permalink:: http://dlmf.nist.gov/21.7.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(ii), §21.7 and Ch.21

… ►

►Let

\Gamma

be a hyperelliptic Riemann surface. …

… ►

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)

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable, $\nu$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $\lambda=-4q$ , $\Psi_{g,N}(\lambda,z)$ : polynomial, $g$ : degree of $\lambda$ and $N$ : degree of $z$
Referenced by:: §31.8
Permalink:: http://dlmf.nist.gov/31.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.8 and Ch.31

… ►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})

. …