# general lemniscatic case

(0.002 seconds)

## 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 Figure 23.4.1: ℘ ⁡ ( x ; g 2 ⁡ , 0 ) for 0 ≤ x ≤ 9 , g 2 ⁡ = 0. …(Lemniscatic case.) Magnify Figure 23.4.3: ζ ⁡ ( x ; g 2 ⁡ , 0 ) for 0 ≤ x ≤ 8 , g 2 ⁡ = 0. …(Lemniscatic case.) Magnify Figure 23.4.5: σ ⁡ ( x ; g 2 ⁡ , 0 ) for - 5 ≤ x ≤ 5 , g 2 ⁡ = 0. …(Lemniscatic case.) Magnify Figure 23.4.7: ℘ ⁡ ( x ) with ω 1 = K ⁡ ( k ) , ω 3 = i ⁢ K ′ ⁡ ( k ) for 0 ≤ x ≤ 9 , k 2 = 0. …(Lemniscatic case.) Magnify
##### 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. …