21 Multidimensional Theta Functions Properties 21.3 Symmetry and Quasi-Periodicity 21.5 Modular Transformations

§21.4 Graphics

ⓘ

Keywords:: Riemann theta functions, graphics
Notes:: These graphics were computed by the author, using the algorithms described in Deconinck et al. (2004).
Permalink:: http://dlmf.nist.gov/21.4
See also:: Annotations for Ch.21

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).

See accompanying text — Figure 21.4.1: $\hat{\theta}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ parametrized by (21.4.1). The surface plots are of $\hat{\theta}\left(x+iy,0\middle|\boldsymbol{{\Omega}}\right)$ , $0\leq x\leq 1$ , $0\leq y\leq 5$ (suffix 1); $\hat{\theta}\left(x,y\middle|\boldsymbol{{\Omega}}\right)$ , $0\leq x\leq 1$ , $0\leq y\leq 1$ (suffix 2); $\hat{\theta}\left(ix,iy\middle|\boldsymbol{{\Omega}}\right)$ , $0\leq x\leq 5$ , $0\leq y\leq 5$ (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 21.4.2–21.4.5

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

21.3 Symmetry and Quasi-Periodicity 21.5 Modular Transformations

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(a₁)	(b₁)	(c₁)

(a₂)	(b₂)	(c₂)

(a₃)	(b₃)	(c₃)