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

§21.4 Graphics

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

Figure 21.4.1 provides surfaces of the scaled Riemann theta function $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}% }\right)$ , 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:: $\boldsymbol{{\Omega}}$ : a Riemann matrix
Referenced by:: Figure 21.4.1, Figure 21.4.1
Permalink:: http://dlmf.nist.gov/21.4.E1
Encodings:: TeX, pMML, png

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).

Visualization Help	Visualization Help	Visualization Help
(a ${}_{1}$ )	(b ${}_{1}$ )	(c ${}_{1}$ )
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(a ${}_{2}$ )	(b ${}_{2}$ )	(c ${}_{2}$ )
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(a ${}_{3}$ )	(b ${}_{3}$ )	(c ${}_{3}$ )

Figure 21.4.1: $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}% }\right)$ parametrized by (21.4.1). The surface plots are of $\mathop{\hat{\theta}\/}\nolimits\!\left(x+iy,0\middle|\boldsymbol{{\Omega}}\right)$ , $0\leq x\leq 1$ , $0\leq y\leq 5$ (suffix 1); $\mathop{\hat{\theta}\/}\nolimits\!\left(x,y\middle|\boldsymbol{{\Omega}}\right)$ , $0\leq x\leq 1$ , $0\leq y\leq 1$ (suffix 2); $\mathop{\hat{\theta}\/}\nolimits\!\left(ix,iy\middle|\boldsymbol{{\Omega}}\right)$ , $0\leq x\leq 5$ , $0\leq y\leq 5$ (suffix 3). Shown are the real part (a), the imaginary part (b), and the modulus (c).

Symbols:: $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}% }\right)$ : scaled Riemann theta function and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Referenced by:: §21.4
Permalink:: http://dlmf.nist.gov/21.4.F1
Encodings:: VRML, VRML, VRML, VRML, VRML, VRML, VRML, VRML, VRML, X3D, X3D, X3D, X3D, X3D, X3D, X3D, X3D, X3D, pdf, pdf, pdf, pdf, pdf, pdf, pdf, pdf, pdf, png, png, png, png, png, png, png, png, png

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:: $\boldsymbol{{\Omega}}$ : a Riemann matrix
Permalink:: http://dlmf.nist.gov/21.4.E2
Encodings:: TeX, pMML, png

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:: $\boldsymbol{{\Omega}}$ : a Riemann matrix
Permalink:: http://dlmf.nist.gov/21.4.E3
Encodings:: TeX, pMML, png

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Figure 21.4.2: $\realpart{\mathop{\hat{\theta}\/}\nolimits\!\left(x+iy,0\middle|\boldsymbol{{% \Omega}}_{1}\right)}$ , $0\leq x\leq 1$ , $0\leq y\leq 5$ . (The imaginary part looks very similar.)

Symbols:: $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}% }\right)$ : scaled Riemann theta function, $\realpart{}$ : real part and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Referenced by:: §21.4
Permalink:: http://dlmf.nist.gov/21.4.F2
Encodings:: VRML, X3D, pdf, png

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Figure 21.4.3: $\left|\mathop{\hat{\theta}\/}\nolimits\!\left(x+iy,0\middle|\boldsymbol{{% \Omega}}_{1}\right)\right|$ , $0\leq x\leq 1$ , $0\leq y\leq 2$ .

Symbols:: $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}% }\right)$ : scaled Riemann theta function and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Permalink:: http://dlmf.nist.gov/21.4.F3
Encodings:: VRML, X3D, pdf, png

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Figure 21.4.4: A real-valued scaled Riemann theta function: $\mathop{\hat{\theta}\/}\nolimits\!\left(ix,iy\middle|\boldsymbol{{\Omega}}_{1}\right)$ , $0\leq x\leq 4$ , $0\leq y\leq 4$ . In this case, the quasi-periods are commensurable, resulting in a doubly-periodic configuration.

Symbols:: $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}% }\right)$ : scaled Riemann theta function and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Permalink:: http://dlmf.nist.gov/21.4.F4
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: $\realpart{\mathop{\hat{\theta}\/}\nolimits\!\left(x+iy,0,0\middle|\boldsymbol{% {\Omega}}_{2}\right)}$ , $0\leq x\leq 1$ , $0\leq y\leq 3$ . This Riemann matrix originates from the genus 3 Riemann surface represented by the algebraic curve $\mu^{3}+2\mu-\lambda^{4}=0$ ; compare §21.7(i).

Symbols:: $\mathop{\hat{\theta}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}% }\right)$ : scaled Riemann theta function, $\realpart{}$ : real part and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Referenced by:: §21.4
Permalink:: http://dlmf.nist.gov/21.4.F5
Encodings:: VRML, X3D, pdf, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 21.3 Symmetry and Quasi-Periodicity 21.5 Modular Transformations