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

§20.3 Graphics

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

§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

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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

Figure 20.3.1: $\theta_{j}\left(\pi x,0.15\right)$ , $0\leq x\leq 2$ ,

. Magnify

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and $\pi$ : the ratio of the circumference of a circle to its diameter
Referenced by:: Erratum (V1.0.16) for Figure 20.3.1
Permalink:: http://dlmf.nist.gov/20.3.F1
Encodings:: pdf, png
Errata (effective with 1.0.16):: The locations of the tick marks denoting 1.5 and 2 on the -axis were corrected.
Reported 2017-05-22 by Paul Abbott
See also:: Annotations for §20.3(i), §20.3 and Ch.20

Figure 20.3.2: $\theta_{1}\left(\pi x,q\right)$ , $0\leq x\leq 2$ ,

= 0.05, 0.5, 0.7, 0.9. For $q\leq q^{\text{Dedekind}}$ , $\theta_{1}\left(\pi x,q\right)$ is convex in

for

. 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

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Symbols:: $\eta\left(\NVar{\tau}\right)$ : Dedekind’s eta function (or Dedekind modular function), $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and : nome
Keywords:: Dedekind’s eta function, modular functions, relations to other functions, relations to theta functions, theta functions
Referenced by:: Figure 23.16.1, Figure 23.16.1
Permalink:: http://dlmf.nist.gov/20.3.F2
Encodings:: pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

Figure 20.3.3: $\theta_{2}\left(\pi x,q\right)$ , $0\leq x\leq 2$ ,

= 0.05, 0.5, 0.7, 0.9. Magnify

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and : nome
Permalink:: http://dlmf.nist.gov/20.3.F3
Encodings:: pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

Figure 20.3.4: $\theta_{3}\left(\pi x,q\right)$ , $0\leq x\leq 2$ ,

= 0.05, 0.5, 0.7, 0.9. Magnify

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and : nome
Permalink:: http://dlmf.nist.gov/20.3.F4
Encodings:: pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

Figure 20.3.5: $\theta_{4}\left(\pi x,q\right)$ , $0\leq x\leq 2$ ,

= 0.05, 0.5, 0.7, 0.9. Magnify

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and : nome
Permalink:: http://dlmf.nist.gov/20.3.F5
Encodings:: pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

Figure 20.3.6: $\theta_{1}\left(x,q\right)$ , $0\leq q\leq 1$ ,

= 0, 0.4, 5, 10, 40. Magnify

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and : nome
Permalink:: http://dlmf.nist.gov/20.3.F6
Encodings:: pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

Figure 20.3.7: $\theta_{2}\left(x,q\right)$ , $0\leq q\leq 1$ ,

= 0, 0.4, 5, 10, 40. Magnify

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and : nome
Permalink:: http://dlmf.nist.gov/20.3.F7
Encodings:: pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

Figure 20.3.8: $\theta_{3}\left(x,q\right)$ , $0\leq q\leq 1$ ,

= 0, 0.4, 5, 10, 40. Magnify

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and : nome
Permalink:: http://dlmf.nist.gov/20.3.F8
Encodings:: pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

Figure 20.3.9: $\theta_{4}\left(x,q\right)$ , $0\leq q\leq 1$ ,

= 0, 0.4, 5, 10, 40. Magnify

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function and : nome
Permalink:: http://dlmf.nist.gov/20.3.F9
Encodings:: pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

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

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and : nome
Permalink:: http://dlmf.nist.gov/20.3.F10
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

Figure 20.3.11: $\theta_{2}\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ . Magnify 3D Help

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and : nome
Permalink:: http://dlmf.nist.gov/20.3.F11
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

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

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and : nome
Permalink:: http://dlmf.nist.gov/20.3.F12
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

Figure 20.3.13: $\theta_{4}\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ . Magnify 3D Help

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and : nome
Permalink:: http://dlmf.nist.gov/20.3.F13
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(i), §20.3 and Ch.20

§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.

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

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/20.3.F14
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(ii), §20.3 and Ch.20

Figure 20.3.15: $\theta_{2}\left(\pi x+iy,0.12\right)$ , $-1\leq x\leq 1$ , $-1\leq y\leq 2.3$ . Magnify 3D Help

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/20.3.F15
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(ii), §20.3 and Ch.20

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

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/20.3.F16
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(ii), §20.3 and Ch.20

Figure 20.3.17: $\theta_{4}\left(\pi x+iy,0.12\right)$ , $-1\leq x\leq 1$ , $-1\leq y\leq 1.5$ . Magnify 3D Help

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/20.3.F17
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(ii), §20.3 and Ch.20

§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.

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

is chosen arbitrarily since $\theta_{1}$ vanishes identically when

. Magnify 3D Help

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\mathrm{i}$ : imaginary unit and : complex
Permalink:: http://dlmf.nist.gov/20.3.F18
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(iii), §20.3 and Ch.20

Figure 20.3.19: $\theta_{2}\left(0\middle|u+iv\right)$ , $-1\leq u\leq 1$ , $0.005\leq v\leq 0.1$ . Magnify 3D Help

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/20.3.F19
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(iii), §20.3 and Ch.20

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

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/20.3.F20
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(iii), §20.3 and Ch.20

Figure 20.3.21: $\theta_{4}\left(0\middle|u+iv\right)$ , $-1\leq u\leq 1$ , $0.005\leq v\leq 0.1$ . Magnify 3D Help

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function and $\mathrm{i}$ : imaginary unit
Permalink:: http://dlmf.nist.gov/20.3.F21
Encodings:: Vizualization, pdf, png
See also:: Annotations for §20.3(iii), §20.3 and Ch.20

20.2 Definitions and Periodic Properties 20.4 Values at

= 0

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§20.3 Graphics

Contents

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

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

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

§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