# general lemniscatic case

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## 4 matching pages

##### 1: 19.20 Special Cases
The general lemniscatic case is … The general lemniscatic case is …
##### 2: 23.22 Methods of Computation
• (b)

If $d=0$, then

23.22.2 $2\omega_{1}=-2i\omega_{3}=\frac{\left(\Gamma\left(\frac{1}{4}\right)\right)^{2% }}{2\sqrt{\pi}c^{1/4}}.$

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 $\omega_{1}>0$.

• ##### 3: 23.4 Graphics
###### §23.4(i) Real Variables
Line graphs of the Weierstrass functions $\wp\left(x\right)$, $\zeta\left(x\right)$, and $\sigma\left(x\right)$, illustrating the lemniscatic and equianharmonic cases. …
##### 4: 23.5 Special Lattices
This happens in the cases treated in the following four subsections. …
###### §23.5(iv) Rhombic Lattice
$e_{1}$ and $g_{3}$ have the same sign unless $2\omega_{3}=(1+i)\omega_{1}$ when both are zero: the pseudo-lemniscatic case. As a function of $\Im e_{3}$ the root $e_{1}$ is increasing. …