DLMF:

Figure 20.3.2 (See in context.)

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.

Annotations:

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
Permalink:: http://dlmf.nist.gov/20.3.F2.mag
Encodings:: Magnified png, pdf