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21 Multidimensional Theta FunctionsProperties

§21.4 Graphics

Figure 21.4.1 provides surfaces of the scaled Riemann theta function θ^(z|Ω), with

21.4.1 Ω=[1.69098 3006+0.95105 6516i1.5+0.36327 1264i1.5+0.36327 1264i1.30901 6994+0.95105 6516i].

This Riemann matrix originates from the Riemann surface represented by the algebraic curve μ3-λ7+2λ3μ=0; compare §21.7(i).

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(a1) (b1) (c1)
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(a2) (b2) (c2)
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(a3) (b3) (c3)
Figure 21.4.1: θ^(z|Ω) parametrized by (21.4.1). The surface plots are of θ^(x+iy,0|Ω), 0x1, 0y5 (suffix 1); θ^(x,y|Ω), 0x1, 0y1 (suffix 2); θ^(ix,iy|Ω), 0x5, 0y5 (suffix 3). Shown are the real part (a), the imaginary part (b), and the modulus (c). Magnify 3D Help

For the scaled Riemann theta functions depicted in Figures 21.4.221.4.5

21.4.2 Ω1=[i-12-12i],

and

21.4.3 Ω2=[-12+i12-12i-12-12i12-12ii0-12-12i0i].
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Figure 21.4.2: θ^(x+iy,0|Ω1), 0x1, 0y5. (The imaginary part looks very similar.) Magnify 3D Help
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Figure 21.4.3: |θ^(x+iy,0|Ω1)|, 0x1, 0y2. Magnify 3D Help
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Figure 21.4.4: A real-valued scaled Riemann theta function: θ^(ix,iy|Ω1), 0x4, 0y4. In this case, the quasi-periods are commensurable, resulting in a doubly-periodic configuration. Magnify 3D Help
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Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: θ^(x+iy,0,0|Ω2), 0x1, 0y3. This Riemann matrix originates from the genus 3 Riemann surface represented by the algebraic curve μ3+2μ-λ4=0; compare §21.7(i). Magnify 3D Help