Weber modular function

Let $q\; =\; e^\{2\backslash pi\; i\; \backslash tau\}$ where τ is an element of the upper half-plane. Then the Weber functions are

:$\backslash begin\{align\}$

\mathfrak{f}(\tau) &= q^{-\frac{1}{48}}\prod_{n>0}(1+q^{n-1/2}) = \frac{\eta^2(\tau)}{\eta\big(\tfrac{\tau}{2}\big)\eta(2\tau)} = e^{-\frac{\pi i}{24}}\frac{\eta\big(\frac{\tau+1}{2}\big)}{\eta(\tau)},\\

\mathfrak{f}_1(\tau) &= q^{-\frac{1}{48}}\prod_{n>0}(1-q^{n-1/2}) = \frac{\eta\big(\tfrac{\tau}{2}\big)}{\eta(\tau)},\\

\mathfrak{f}_2(\tau) &= \sqrt2\, q^{\frac{1}{24}}\prod_{n>0}(1+q^{n})= \frac{\sqrt2\,\eta(2\tau)}{\eta(\tau)}.

\end{align}

These are also the definitions in Duke's paper "Continued Fractions and Modular Functions".