Aquileo | DLMF: Figure 20.3.2 ‣ §20.3 Graphics ‣ Properties ‣ Chapter 20 Theta Functions

Figure 20.3.2 (See in context.)

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.

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Annotations:

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
Permalink:: http://dlmf.nist.gov/20.3.F2.mag
Encodings:: Magnified png, pdf
See also:: Annotations for §20.3 and Ch.20