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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.

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1 Answer 1

up vote 7 down vote accepted

From equation (II) in Newman's A simple proof of the partition formula:

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

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 $z \in [0,1)$.

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