# §21.4 Graphics

Figure 21.4.1 provides surfaces of the scaled Riemann theta function $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$, with

 21.4.1 $\boldsymbol{{\Omega}}=\begin{bmatrix}1.69098\;3006+0.95105\;6516\,i&1.5+0.3632% 7\;1264\,i\\ 1.5+0.36327\;1264\,i&1.30901\;6994+0.95105\;6516\,i\end{bmatrix}.$ ⓘ Symbols: $\mathrm{i}$: imaginary unit and $\boldsymbol{{\Omega}}$: a Riemann matrix Referenced by: Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1 Permalink: http://dlmf.nist.gov/21.4.E1 Encodings: TeX, pMML, png See also: Annotations for §21.4 and Ch.21

This Riemann matrix originates from the Riemann surface represented by the algebraic curve $\mu^{3}-\lambda^{7}+2\lambda^{3}\mu=0$; compare §21.7(i). Figure 21.4.1: θ^⁡(𝐳|𝛀) parametrized by (21.4.1). The surface plots are of θ^⁡(x+i⁢y,0|𝛀), 0≤x≤1, 0≤y≤5 (suffix 1); θ^⁡(x,y|𝛀), 0≤x≤1, 0≤y≤1 (suffix 2); θ^⁡(i⁢x,i⁢y|𝛀), 0≤x≤5, 0≤y≤5 (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 $\boldsymbol{{\Omega}}_{1}=\begin{bmatrix}i&-\tfrac{1}{2}\\ -\tfrac{1}{2}&i\end{bmatrix},$ ⓘ Symbols: $\mathrm{i}$: imaginary unit and $\boldsymbol{{\Omega}}$: a Riemann matrix Permalink: http://dlmf.nist.gov/21.4.E2 Encodings: TeX, pMML, png See also: Annotations for §21.4 and Ch.21

and

 21.4.3 $\boldsymbol{{\Omega}}_{2}=\begin{bmatrix}-\tfrac{1}{2}+i&\tfrac{1}{2}-\tfrac{1% }{2}i&-\tfrac{1}{2}-\tfrac{1}{2}i\\ \tfrac{1}{2}-\tfrac{1}{2}i&i&0\\ -\tfrac{1}{2}-\tfrac{1}{2}i&0&i\end{bmatrix}.$ ⓘ Symbols: $\mathrm{i}$: imaginary unit and $\boldsymbol{{\Omega}}$: a Riemann matrix Permalink: http://dlmf.nist.gov/21.4.E3 Encodings: TeX, pMML, png See also: Annotations for §21.4 and Ch.21 Figure 21.4.2: ℜ⁡θ^⁡(x+i⁢y,0|𝛀1), 0≤x≤1, 0≤y≤5. (The imaginary part looks very similar.) Magnify 3D Help Figure 21.4.4: A real-valued scaled Riemann theta function: θ^⁡(i⁢x,i⁢y|𝛀1), 0≤x≤4, 0≤y≤4. In this case, the quasi-periods are commensurable, resulting in a doubly-periodic configuration. Magnify 3D Help