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22 Jacobian Elliptic FunctionsProperties

§22.3 Graphics

Contents

§22.3(i) Real Variables: Line Graphs

Line graphs of the functions sn(x,k), cn(x,k), dn(x,k), cd(x,k), sd(x,k), nd(x,k), dc(x,k), nc(x,k), sc(x,k), ns(x,k), ds(x,k), and cs(x,k) for representative values of real x and real k illustrating the near trigonometric (k=0), and near hyperbolic (k=1) limits.

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Figure 22.3.1: k=0.4, -3Kx3K, K=1.6399. Magnify
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Figure 22.3.2: k=0.7, -3Kx3K, K=1.8456. For cn(x,k) the curve for k=1/2=0.70710 is a boundary between the curves that have an inflection point in the interval 0x2K(k), and its translates, and those that do not; see Walker (1996, p. 146). Magnify
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Figure 22.3.3: k=0.99, -3Kx3K, K=3.3566. Magnify
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Figure 22.3.4: k=0.999999, -3Kx3K, K=7.9474. Magnify
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Figure 22.3.5: k=0.4, -2Kx2K, K=1.6399. Magnify
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Figure 22.3.6: k=0.7, -2Kx2K, K=1.8456. Magnify
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Figure 22.3.7: k=0.99, -2Kx2K, K=3.3566. Magnify
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Figure 22.3.8: k=0.999999, -2Kx2K, K=7.9474. Magnify
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Figure 22.3.9: k=0.4, -2Kx2K, K=1.6399. Magnify
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Figure 22.3.10: k=0.7, -2Kx2K, K=1.8456. Magnify
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Figure 22.3.11: k=0.99, -2Kx2K, K=3.3566. Magnify
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Figure 22.3.12: k=0.999999, -2Kx2K, K=7.9474. Magnify

§22.3(ii) Real Variables: Surfaces

sn(x,k), cn(x,k), and dn(x,k) as functions of real arguments x and k. The period diverges logarithmically as k1-; see §19.12.

Figure 22.3.13: sn(x,k) for k=1-e-n, n=0 to 20, -5πx5π. Magnify
Figure 22.3.14: cn(x,k) for k=1-e-n, n=0 to 20, -5πx5π. Magnify
Figure 22.3.15: dn(x,k) for k=1-e-n, n=0 to 20, -5πx5π. Magnify

§22.3(iii) Complex z; Real k

In the graphics shown in this subsection height corresponds to the absolute value of the function and color to the phase. See About Color Map.

Figure 22.3.16: sn(x+iy,k) for k=0.99, -3Kx3K, 0y4K. K=3.3566, K=1.5786. Magnify
Figure 22.3.17: cn(x+iy,k) for k=0.99, -3Kx3K, 0y4K. K=3.3566, K=1.5786. Magnify
Figure 22.3.18: dn(x+iy,k) for k=0.99, -3Kx3K, 0y4K. K=3.3566, K=1.5786. Magnify
Figure 22.3.19: cd(x+iy,k) for k=0.99, -3Kx3K, 0y4K. K=3.3566, K=1.5786. Magnify
Figure 22.3.20: dc(x+iy,k) for k=0.99, -3Kx3K, 0y4K. K=3.3566, K=1.5786. Magnify
Figure 22.3.21: ns(x+iy,k) for k=0.99, -3Kx3K, 0y4K. K=3.3566, K=1.5786. Magnify

§22.3(iv) Complex k

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Figure 22.3.22: sn(x,k), x=120, as a function of k2=iκ2, 0κ4. Magnify
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Figure 22.3.23: sn(x,k), x=120, as a function of k2=iκ2, 0κ4. Magnify

In Figures 22.3.24 and 22.3.25, height corresponds to the absolute value of the function and color to the phase. See p. About Color Map.

Figure 22.3.24: sn(x+iy,k) for -4x4, 0y8, k=1+12i. K=1.5149+i0.5235, K=1.4620-i0.3552. Magnify
Figure 22.3.25: sn(5,k) as a function of complex k2, -1(k2)3.5, -1(k2)1. Compare §22.17(ii). Magnify
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Figure 22.3.26: Density plot of |sn(5,k)| as a function of complex k2, -10(k2)20, -10(k2)10. Grayscale, running from 0 (black) to 10 (white), with |(sn(5,k))|>10 truncated to 10. White spots correspond to poles. Magnify
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Figure 22.3.27: Density plot of |sn(10,k)| as a function of complex k2, -10(k2)20, -10(k2)10. Grayscale, running from 0 (black) to 10 (white), with |sn(10,k)|>10 truncated to 10. White spots correspond to poles. Magnify
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Figure 22.3.28: Density plot of |sn(20,k)| as a function of complex k2, -10(k2)20, -10(k2)10. Grayscale, running from 0 (black) to 10 (white), with |sn(20,k)|>10 truncated to 10. White spots correspond to poles. Magnify
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Figure 22.3.29: Density plot of |sn(30,k)| as a function of complex k2, -10(k2)20, -10(k2)10. Grayscale, running from 0 (black) to 10 (white), with |sn(30,k)|>10 truncated to 10. White spots correspond to poles. Magnify