general lemniscatic case

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1: 19.20 Special Cases

… ►The general lemniscatic case is … ►The general lemniscatic case is …

2: 23.22 Methods of Computation

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If $d=0$, then

23.22.2 $$2{\omega }_1=-2\mathrm{i}{\omega }_3=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{4}\right)\right)}^2}{2\sqrt{\pi }{c}^{1/4}}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators

Referenced by:

§22.20(iv), §23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

There are 4 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 4 rotations of a square lattice. The lemniscatic case occurs when $c>0$ and ${\omega }_1>0$.

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3: 23.4 Graphics

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§23.4(i) Real Variables

►Line graphs of the Weierstrass functions $\mathrm{\wp }\left(x\right)$, $\zeta \left(x\right)$, and $\sigma \left(x\right)$, illustrating the lemniscatic and equianharmonic cases. … ►

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Figure 23.4.1: $\mathrm{\wp }(x;g_2,0)$ for $0\le x\le 9$, $g_2$ = 0. …(Lemniscatic case.) Magnify

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Figure 23.4.3: $\zeta (x;g_2,0)$ for $0\le x\le 8$, $g_2$ = 0. …(Lemniscatic case.) Magnify

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Figure 23.4.5: $\sigma (x;g_2,0)$ for $-5\le x\le 5$, $g_2$ = 0. …(Lemniscatic case.) Magnify

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4: 23.5 Special Lattices

… ►This happens in the cases treated in the following four subsections. … ►

§23.5(iii) Lemniscatic Lattice

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§23.5(iv) Rhombic Lattice

… ► $e_1$ and $g_3$ have the same sign unless $2{\omega }_3=(1+\mathrm{i}){\omega }_1$ when both are zero: the pseudo-lemniscatic case. As a function of $\mathrm{\Im }{e}_3$ the root $e_1$ is increasing. …