Anybody knows how to compute the Jacobi triple product accurately as $x \rightarrow 1$:
$$ \prod_{m=1}^{\infty}(1-x^{2m})(1 + x^{2m-1}y^{-2})(1 + x^{2m-1}y^{2}) = \sum_{n=-\infty}^{\infty}x^{n^2}y^{-2n} $$
From the right hand series there is a singularity for $x=1$. Is there any series that isolates the singularity there and is accurate with a few terms ?
Your expression $$ \prod_{m=1}^{\infty} (1\!-\!x^{2m})(1\!+\!x^{2m-1}y^{-2}) (1\!+\!x^{2m-1}y^{2})\!=\!\sum_{n=-\infty}^{\infty}x^{n^2}y^{-2n}$$ is a Jacobi theta function according to DLMF equation 20.5.3 which is equal to $\,\theta_3(z,q)\,$ where $\,x = q\,$ and $\, y = \exp(i\,z).\,$ Your question about $\,q\to 1\,$ is addressed in DLMF equation 20.7.32 which is $$ (\tau/i)^{1/2} \theta_3(z|\tau) = \exp(i\,\tau'z^2/\pi)\, \theta_3(z\,\tau'|\tau') $$ where $\,q = e^{\pi i \tau}\,$ since as $\,\tau\to 0\,$ we have $\,\tau'\to i\infty\,$ or $\,q'\to 0.$
