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20 Theta Functions Properties 20.2 Definitions and Periodic Properties 20.4 Values at

z

§20.3 Graphics

Permalink:: http://dlmf.nist.gov/20.3
See also:: Annotations for 20

Contents

§20.3(i) $\theta$ -Functions: Real Variable and Real Nome
§20.3(ii) $\theta$ -Functions: Complex Variable and Real Nome
§20.3(iii) $\theta$ -Functions: Real Variable and Complex Lattice Parameter

§20.3(i) $\theta$ -Functions: Real Variable and Real Nome

Keywords:: theta functions
Notes:: These graphics were produced at NIST.
Permalink:: http://dlmf.nist.gov/20.3.i
See also:: Annotations for 20.3

See accompanying text — Figure 20.3.1: $\mathop{\theta_{j}\/}\nolimits\!\left(\pi x,0.15\right)$ , $0\leq x\leq 2$ , $j=1,2,3,4$ .

Magnify — Figure 20.3.1: $\mathop{\theta_{j}\/}\nolimits\!\left(\pi x,0.15\right)$ , $0\leq x\leq 2$ , $j=1,2,3,4$ .

See accompanying text — Figure 20.3.2: $\mathop{\theta_{1}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $q$ = 0.05, 0.5, 0.7, 0.9. For $q\leq q^{\text{Dedekind}}$ , $\mathop{\theta_{1}\/}\nolimits\!\left(\pi x,q\right)$ is convex in $x$ for $0<x<1$ . Here $q^{\text{Dedekind}}=e^{-\pi y_{0}}=0.19$ approximately, where $y=y_{0}$ corresponds to the maximum value of Dedekind’s eta function $\mathop{\eta\/}\nolimits\!\left(iy\right)$ as depicted in Figure 23.16.1.

Magnify — Figure 20.3.2: $\mathop{\theta_{1}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $q$ = 0.05, 0.5, 0.7, 0.9. For $q\leq q^{\text{Dedekind}}$ , $\mathop{\theta_{1}\/}\nolimits\!\left(\pi x,q\right)$ is convex in $x$ for $0<x<1$ . Here $q^{\text{Dedekind}}=e^{-\pi y_{0}}=0.19$ approximately, where $y=y_{0}$ corresponds to the maximum value of Dedekind’s eta function $\mathop{\eta\/}\nolimits\!\left(iy\right)$ as depicted in Figure 23.16.1.

See accompanying text — Figure 20.3.4: $\mathop{\theta_{3}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $q$ = 0.05, 0.5, 0.7, 0.9.

Magnify — Figure 20.3.4: $\mathop{\theta_{3}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $q$ = 0.05, 0.5, 0.7, 0.9.

See accompanying text — Figure 20.3.6: $\mathop{\theta_{1}\/}\nolimits\!\left(x,q\right)$ , $0\leq q\leq 1$ , $x$ = 0, 0.4, 5, 10, 40.

Magnify — Figure 20.3.6: $\mathop{\theta_{1}\/}\nolimits\!\left(x,q\right)$ , $0\leq q\leq 1$ , $x$ = 0, 0.4, 5, 10, 40.

See accompanying text — Figure 20.3.8: $\mathop{\theta_{3}\/}\nolimits\!\left(x,q\right)$ , $0\leq q\leq 1$ , $x$ = 0, 0.4, 5, 10, 40.

Magnify — Figure 20.3.8: $\mathop{\theta_{3}\/}\nolimits\!\left(x,q\right)$ , $0\leq q\leq 1$ , $x$ = 0, 0.4, 5, 10, 40.

See accompanying text — Figure 20.3.10: $\mathop{\theta_{1}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ .

Magnify — Figure 20.3.10: $\mathop{\theta_{1}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ .

See accompanying text — Figure 20.3.12: $\mathop{\theta_{3}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ .

Magnify — Figure 20.3.12: $\mathop{\theta_{3}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ .

§20.3(ii) $\theta$ -Functions: Complex Variable and Real Nome

Keywords:: theta functions
Notes:: These surfaces were produced at NIST.
Permalink:: http://dlmf.nist.gov/20.3.ii
See also:: Annotations for 20.3

In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. See also About Color Map.

See accompanying text — Figure 20.3.14: $\mathop{\theta_{1}\/}\nolimits\!\left(\pi x+iy,0.12\right)$ , $-1\leq x\leq 1$ , $-1\leq y\leq 2.3$ .

Magnify — Figure 20.3.14: $\mathop{\theta_{1}\/}\nolimits\!\left(\pi x+iy,0.12\right)$ , $-1\leq x\leq 1$ , $-1\leq y\leq 2.3$ .

See accompanying text — Figure 20.3.16: $\mathop{\theta_{3}\/}\nolimits\!\left(\pi x+iy,0.12\right)$ , $-1\leq x\leq 1$ , $-1\leq y\leq 1.5$ .

Magnify — Figure 20.3.16: $\mathop{\theta_{3}\/}\nolimits\!\left(\pi x+iy,0.12\right)$ , $-1\leq x\leq 1$ , $-1\leq y\leq 1.5$ .

§20.3(iii) $\theta$ -Functions: Real Variable and Complex Lattice Parameter

Keywords:: theta functions
Notes:: These surfaces were produced at NIST.
Permalink:: http://dlmf.nist.gov/20.3.iii
See also:: Annotations for 20.3

In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. See also About Color Map.

See accompanying text — Figure 20.3.18: $\mathop{\theta_{1}\/}\nolimits\!\left(0.1\middle|u+iv\right)$ , $-1\leq u\leq 1$ , $0.005\leq v\leq 0.5$ . The value 0.1 of $z$ is chosen arbitrarily since $\mathop{\theta_{1}\/}\nolimits$ vanishes identically when $z=0$ .

Magnify — Figure 20.3.18: $\mathop{\theta_{1}\/}\nolimits\!\left(0.1\middle|u+iv\right)$ , $-1\leq u\leq 1$ , $0.005\leq v\leq 0.5$ . The value 0.1 of $z$ is chosen arbitrarily since $\mathop{\theta_{1}\/}\nolimits$ vanishes identically when $z=0$ .

See accompanying text — Figure 20.3.20: $\mathop{\theta_{3}\/}\nolimits\!\left(0\middle|u+iv\right)$ , $-1\leq u\leq 1$ , $0.005\leq v\leq 0.1$ .

Magnify — Figure 20.3.20: $\mathop{\theta_{3}\/}\nolimits\!\left(0\middle|u+iv\right)$ , $-1\leq u\leq 1$ , $0.005\leq v\leq 0.1$ .

© 2010–2016 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.14; Release date 2016-12-21. A printed companion is available. 20.2 Definitions and Periodic Properties 20.4 Values at

z