equianharmonic

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1: 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 23.4.2: $\mathrm{\wp }(x;0,g_3)$ for $0\le x\le 9$, $g_3$ = 0. …(Equianharmonic case.) Magnify

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Figure 23.4.4: $\zeta (x;0,g_3)$ for $0\le x\le 8$, $g_3$ = 0. …(Equianharmonic case.) Magnify

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Figure 23.4.6: $\sigma (x;0,g_3)$ for $-5\le x\le 5$, $g_3$ = 0. …(Equianharmonic case.) Magnify

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2: 23.5 Special Lattices

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§23.5(v) Equianharmonic Lattice

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Figure 23.5.2: Equianharmonic lattice. … Magnify

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3: 22.5 Special Values

… ►For values of $K,K^{\prime }$ when $k^2=\frac{1}{2}$ (lemniscatic case) see §23.5(iii), and for $k^2={\mathrm{e}}^{\mathrm{i}\pi /3}$ (equianharmonic case) see §23.5(v).

4: 23.22 Methods of Computation

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If $c=0$, then

23.22.3 $$2{\omega }_1=2{\mathrm{e}}^{-\pi \mathrm{i}/3}{\omega }_3=\frac{{\left(\mathrm{\Gamma }\left(\frac{1}{3}\right)\right)}^3}{2\pi d^{1/6}}.$$

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Symbols:

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

Referenced by:

§23.22(ii)

Permalink:

http://dlmf.nist.gov/23.22.E3

Encodings:

pMML, png, TeX

See also:

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

There are 6 possible pairs ($2{\omega }_1$, $2{\omega }_3$), corresponding to the 6 rotations of a lattice of equilateral triangles. The equianharmonic case occurs when $d>0$ and ${\omega }_1>0$.

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