§20.3 Graphics

Permalink:: http://dlmf.nist.gov/20.3

§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

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

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

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function
Permalink:: http://dlmf.nist.gov/20.3.F1
Encodings:: pdf, png

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.

Symbols:: $\mathop{\eta\/}\nolimits\!\left(\tau\right)$ : Dedekind’s eta function (or Dedekind modular function), $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function, $e$ : base of exponential function and $q$ : nome
Keywords:: Dedekind’s eta function, modular functions, theta functions
Referenced by:: Figure 23.16.1, Figure 23.16.1
Permalink:: http://dlmf.nist.gov/20.3.F2
Encodings:: pdf, png

Figure 20.3.3: $\mathop{\theta _{{2}}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $q$ = 0.05, 0.5, 0.7, 0.9.

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.3.F3
Encodings:: pdf, png

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.

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.3.F4
Encodings:: pdf, png

Figure 20.3.5: $\mathop{\theta _{{4}}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $q$ = 0.05, 0.5, 0.7, 0.9.

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.3.F5
Encodings:: pdf, png

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

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.3.F6
Encodings:: pdf, png

Figure 20.3.7: $\mathop{\theta _{{2}}\/}\nolimits\!\left(x,q\right)$ , $0\leq q\leq 1$ , $x$ = 0, 0.4, 5, 10, 40.

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.3.F7
Encodings:: pdf, png

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

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.3.F8
Encodings:: pdf, png

Figure 20.3.9: $\mathop{\theta _{{4}}\/}\nolimits\!\left(x,q\right)$ , $0\leq q\leq 1$ , $x$ = 0, 0.4, 5, 10, 40.

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.3.F9
Encodings:: pdf, png

Visualization Help

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

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.3.F10
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 20.3.11: $\mathop{\theta _{{2}}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ .

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.3.F11
Encodings:: VRML, X3D, pdf, png

Visualization Help

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

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.3.F12
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 20.3.13: $\mathop{\theta _{{4}}\/}\nolimits\!\left(\pi x,q\right)$ , $0\leq x\leq 2$ , $0\leq q\leq 0.99$ .

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function and $q$ : nome
Permalink:: http://dlmf.nist.gov/20.3.F13
Encodings:: VRML, X3D, pdf, png

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

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

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.

Visualization Help

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

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function
Permalink:: http://dlmf.nist.gov/20.3.F14
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 20.3.15: $\mathop{\theta _{{2}}\/}\nolimits\!\left(\pi x+iy,0.12\right)$ , $-1\leq x\leq 1$ , $-1\leq y\leq 2.3$ .

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function
Permalink:: http://dlmf.nist.gov/20.3.F15
Encodings:: VRML, X3D, pdf, png

Visualization Help

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

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function
Permalink:: http://dlmf.nist.gov/20.3.F16
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 20.3.17: $\mathop{\theta _{{4}}\/}\nolimits\!\left(\pi x+iy,0.12\right)$ , $-1\leq x\leq 1$ , $-1\leq y\leq 1.5$ .

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ : theta function
Permalink:: http://dlmf.nist.gov/20.3.F17
Encodings:: VRML, X3D, pdf, png

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

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

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.

Visualization Help

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 $z=0$ .