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20 Theta FunctionsProperties

§20.3 Graphics

Contents

§20.3(i) θ-Functions: Real Variable and Real Nome

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Figure 20.3.1: θj(πx,0.15), 0x2, j=1,2,3,4. Magnify
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Figure 20.3.2: θ1(πx,q), 0x2, q = 0.05, 0.5, 0.7, 0.9. For qqDedekind, θ1(πx,q) is convex in x for 0<x<1. Here qDedekind=e-πy0=0.19 approximately, where y=y0 corresponds to the maximum value of Dedekind’s eta function η(iy) as depicted in Figure 23.16.1. Magnify
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Figure 20.3.3: θ2(πx,q), 0x2, q = 0.05, 0.5, 0.7, 0.9. Magnify
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Figure 20.3.4: θ3(πx,q), 0x2, q = 0.05, 0.5, 0.7, 0.9. Magnify
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Figure 20.3.5: θ4(πx,q), 0x2, q = 0.05, 0.5, 0.7, 0.9. Magnify
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Figure 20.3.6: θ1(x,q), 0q1, x = 0, 0.4, 5, 10, 40. Magnify
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Figure 20.3.7: θ2(x,q), 0q1, x = 0, 0.4, 5, 10, 40. Magnify
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Figure 20.3.8: θ3(x,q), 0q1, x = 0, 0.4, 5, 10, 40. Magnify
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Figure 20.3.9: θ4(x,q), 0q1, x = 0, 0.4, 5, 10, 40. Magnify
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Figure 20.3.10: θ1(πx,q), 0x2, 0q0.99. Magnify 3D Help
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Figure 20.3.11: θ2(πx,q), 0x2, 0q0.99. Magnify 3D Help
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Figure 20.3.12: θ3(πx,q), 0x2, 0q0.99. Magnify 3D Help
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Figure 20.3.13: θ4(πx,q), 0x2, 0q0.99. Magnify 3D Help

§20.3(ii) θ-Functions: Complex Variable and Real Nome

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

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Figure 20.3.14: θ1(πx+iy,0.12), -1x1, -1y2.3. Magnify 3D Help
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Figure 20.3.15: θ2(πx+iy,0.12), -1x1, -1y2.3. Magnify 3D Help
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Figure 20.3.16: θ3(πx+iy,0.12), -1x1, -1y1.5. Magnify 3D Help
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Figure 20.3.17: θ4(πx+iy,0.12), -1x1, -1y1.5. Magnify 3D Help

§20.3(iii) θ-Functions: Real Variable and Complex Lattice Parameter

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

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Figure 20.3.18: θ1(0.1|u+iv), -1u1, 0.005v0.5. The value 0.1 of z is chosen arbitrarily since θ1 vanishes identically when z=0. Magnify 3D Help
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Figure 20.3.19: θ2(0|u+iv), -1u1, 0.005v0.1. Magnify 3D Help
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Figure 20.3.20: θ3(0|u+iv), -1u1, 0.005v0.1. Magnify 3D Help
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Figure 20.3.21: θ4(0|u+iv), -1u1, 0.005v0.1. Magnify 3D Help