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DLMF: §20.3 Graphics ‣ Properties ‣ Chapter 20 Theta Functions

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

z

§20.3 Graphics

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

Contents

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

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Keywords:: graphics, real variables, theta functions
Notes:: These graphics were produced at NIST.
Permalink:: http://dlmf.nist.gov/20.3.i
See also:: Annotations for §20.3 and Ch.20

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

See accompanying text — Figure 20.3.2: $\theta_{1}\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $q$ = 0.05, 0.5, 0.7, 0.9. For $q\leq q^{\text{Dedekind}}$ , $\theta_{1}\left(\pi x,q\right)$ 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 $\eta\left(iy\right)$ as depicted in Figure 23.16.1. Magnify

See accompanying text — Figure 20.3.2: $\theta_{1}\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $q$ = 0.05, 0.5, 0.7, 0.9. For $q\leq q^{\text{Dedekind}}$ , $\theta_{1}\left(\pi x,q\right)$ 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 $\eta\left(iy\right)$ as depicted in Figure 23.16.1. Magnify

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

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

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

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

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

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

See accompanying text — Figure 20.3.10: $\theta_{1}\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ . Magnify 3D Help

See accompanying text — Figure 20.3.10: $\theta_{1}\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ . Magnify 3D Help

See accompanying text — Figure 20.3.12: $\theta_{3}\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ . Magnify 3D Help

See accompanying text — Figure 20.3.12: $\theta_{3}\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ . Magnify 3D Help

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

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Keywords:: complex variables, graphics, theta functions
Notes:: These surfaces were produced at NIST.
Permalink:: http://dlmf.nist.gov/20.3.ii
See also:: Annotations for §20.3 and Ch.20

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: $\theta_{1}\left(\pi x+iy,0.12\right)$ , $-1\leq x\leq 1$ , $-1\leq y\leq 2.3$ . Magnify 3D Help

See accompanying text — Figure 20.3.14: $\theta_{1}\left(\pi x+iy,0.12\right)$ , $-1\leq x\leq 1$ , $-1\leq y\leq 2.3$ . Magnify 3D Help

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

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

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

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Keywords:: complex variables, graphics, theta functions
Notes:: These surfaces were produced at NIST.
Permalink:: http://dlmf.nist.gov/20.3.iii
See also:: Annotations for §20.3 and Ch.20

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: $\theta_{1}\left(0.1\middle|u+iv\right)$ , $-1\leq u\leq 1$ , $0.005\leq v\leq 0.5$ . The value 0.1 of $z$ is chosen arbitrarily since $\theta_{1}$ vanishes identically when $z=0$ . Magnify 3D Help

See accompanying text — Figure 20.3.18: $\theta_{1}\left(0.1\middle|u+iv\right)$ , $-1\leq u\leq 1$ , $0.005\leq v\leq 0.5$ . The value 0.1 of $z$ is chosen arbitrarily since $\theta_{1}$ vanishes identically when $z=0$ . Magnify 3D Help

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

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

20.2 Definitions and Periodic Properties 20.4 Values at

z

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