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23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions

§23.4 Graphics

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Permalink:
http://dlmf.nist.gov/23.4
See also:
Annotations for Ch.23
Contents
  1. §23.4(i) Real Variables
  2. §23.4(ii) Complex Variables

§23.4(i) Real Variables

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Keywords:
Weierstrass elliptic functions, equianharmonic case, graphics, lattice, lemniscatic, lemniscatic case, real variables
Notes:
These graphs were produced at NIST.
Permalink:
http://dlmf.nist.gov/23.4.i
See also:
Annotations for §23.4 and Ch.23

Line graphs of the Weierstrass functions ℘⁡(x), ζ⁡(x), and σ⁡(x), illustrating the lemniscatic and equianharmonic cases. (The figures in this subsection may be compared with the figures in §22.3(i).)

See accompanying text
Figure 23.4.1: ℘⁡(x;g2⁡,0) for 0≤x≤9, g2⁡ = 0.1, 0.2, 0.5, 0.8. (Lemniscatic case.) Magnify
See accompanying text
Figure 23.4.2: ℘⁡(x;0,g3⁡) for 0≤x≤9, g3⁡ = 0.1, 0.2, 0.5, 0.8. (Equianharmonic case.) Magnify
See accompanying text
Figure 23.4.3: ζ⁡(x;g2⁡,0) for 0≤x≤8, g2⁡ = 0.1, 0.2, 0.5, 0.8. (Lemniscatic case.) Magnify
See accompanying text
Figure 23.4.4: ζ⁡(x;0,g3⁡) for 0≤x≤8, g3⁡ = 0.1, 0.2, 0.5, 0.8. (Equianharmonic case.) Magnify
See accompanying text
Figure 23.4.5: σ⁡(x;g2⁡,0) for −5≤x≤5, g2⁡ = 0.1, 0.2, 0.5, 0.8. (Lemniscatic case.) Magnify
See accompanying text
Figure 23.4.6: σ⁡(x;0,g3⁡) for −5≤x≤5, g3⁡ = 0.1, 0.2, 0.5, 0.8. (Equianharmonic case.) Magnify
See accompanying text
Figure 23.4.7: ℘⁡(x) with ω1=K⁡(k), ω3=i⁢K′⁡(k) for 0≤x≤9, k2 = 0.2, 0.8, 0.95, 0.99. (Lemniscatic case.) Magnify

§23.4(ii) Complex Variables

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Keywords:
Weierstrass elliptic functions, complex variables, graphics
Notes:
These graphics were produced at NIST.
Permalink:
http://dlmf.nist.gov/23.4.ii
See also:
Annotations for §23.4 and Ch.23

Surfaces for the Weierstrass functions ℘⁡(z), ζ⁡(z), and σ⁡(z). Height corresponds to the absolute value of the function and color to the phase. See also About Color Map. (The figures in this subsection may be compared with the figures in §22.3(iii).)

See accompanying text
Figure 23.4.8: ℘⁡(x+i⁢y) with ω1=K⁡(k), ω3=i⁢K′⁡(k) for −2⁢K⁡(k)≤x≤2⁢K⁡(k), 0≤y≤6⁢K′⁡(k), k2=0.9. (The scaling makes the lattice appear to be square.) Magnify 3D Help
See accompanying text
Figure 23.4.9: ℘⁡(x+i⁢y;1,4⁢i) for −3.8≤x≤3.8, −3.8≤y≤3.8. (The variables are unscaled and the lattice is skew.) Magnify 3D Help
See accompanying text
Figure 23.4.10: ζ⁡(x+i⁢y;1,0) for −5≤x≤5, −5≤y≤5. Magnify 3D Help
See accompanying text
Figure 23.4.11: σ⁡(x+i⁢y;1,i) for −2.5≤x≤2.5, −2.5≤y≤2.5. Magnify 3D Help
See accompanying text
Figure 23.4.12: ℘⁡(3.7;a+i⁢b,0) for −5≤a≤3, −4≤b≤4. There is a double zero at a=b=0 and double poles on the real axis. Magnify 3D Help
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