Consider the function


$$f(x)=\prod\limits\_{n=1}^{\infty}(1-x^n)$$
I am interested in an asymptotic formula for $f$ as $x\to 1$. Of course $f\to 0$ but I am interested in how fast.



Growth rate of eta related function
From equation (II) in Newman's [A simple proof of the partition formula](http://projecteuclid.org/euclid.mmj/1028998729):


$$f(z) = \sqrt{\frac{2 \pi}{1-z}} \exp\left(-\frac{\pi^2}{6(1-z)}+\frac{\pi^2}{12} \right) (1+O(1-z))$$


Note that Newman's $f$ is the reciprocal of yours; I rewrote his formula to use your $f$.


This bound is inside the egg shaped region $2 |z| + |1-z| < 2$, and in particular for $z \in [0,1)$. 


