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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
  • (b)

    If d = 0 , then

    23.22.2 2 ω 1 = - 2 i ω 3 = ( Γ ( 1 4 ) ) 2 2 π c 1 / 4 .

    There are 4 possible pairs ( 2 ω 1 , 2 ω 3 ), corresponding to the 4 rotations of a square lattice. The lemniscatic case occurs when c > 0 and ω 1 > 0 .

  • 3: 23.4 Graphics
    §23.4(i) Real Variables
    See accompanying text
    Figure 23.4.1: ( x ; g 2 , 0 ) for 0 x 9 , g 2 = 0. …(Lemniscatic case.) Magnify
    See accompanying text
    Figure 23.4.3: ζ ( x ; g 2 , 0 ) for 0 x 8 , g 2 = 0. …(Lemniscatic case.) Magnify
    See accompanying text
    Figure 23.4.5: σ ( x ; g 2 , 0 ) for - 5 x 5 , g 2 = 0. …(Lemniscatic case.) Magnify
    See accompanying text
    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(iii) Lemniscatic Lattice
    §23.5(iv) Rhombic Lattice
    e 1 and g 3 have the same sign unless 2 ω 3 = ( 1 + i ) ω 1 when both are zero: the pseudo-lemniscatic case. As a function of e 3 the root e 1 is increasing. …