Digital Library of Mathematical Functions
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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 65161.5+0.36327 12641.5+0.36327 12641.30901 6994+0.95105 6516].

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

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

For the scaled Riemann theta functions depicted in Figures 21.4.221.4.5

21.4.2 Ω1=[-12-12],

and

21.4.3 Ω2=[-12+12-12-12-1212-120-12-120].
Figure 21.4.2: θ^(x+y,0|Ω1), 0x1, 0y5. (The imaginary part looks very similar.) Magnify
Figure 21.4.3: |θ^(x+y,0|Ω1)|, 0x1, 0y2. Magnify
Figure 21.4.4: A real-valued scaled Riemann theta function: θ^(x,y|Ω1), 0x4, 0y4. In this case, the quasi-periods are commensurable, resulting in a doubly-periodic configuration. Magnify
Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: θ^(x+y,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