Define \(\displaystyle \mathfrak{f}(\tau) := \prod_{n \geq 1}\left( 1+e^{(2n-1)i\pi \tau}\right)\)

Show that \(\displaystyle \mathfrak{f}(\sqrt{-19})=\theta\) where \(\theta\) is the unique real root of \[x^3-2x-2=0\]

Explicitly,

\(\displaystyle \prod_{n \geq 1}\left( 1+e^{-(2n-1)\pi \sqrt{19}}\right)=\frac{1}{3} \left(27-3 \sqrt{57}\right)^{1/3}+\frac{\left(9+\sqrt{57}\right)^{1/3}}{3^{2/3}}\)