Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2002
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/math/0212035 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- I present and analyze a quadratically convergent algorithm for computing the infinite product \prod_{n=1}^\infty (1 - tx^n) for arbitrary complex t and x satisfying |x| < 1, based on the identity \prod_{n=1}^\infty (1 - tx^n) = \sum_{m=0}^\infty {(-t)^m x^{m(m+1)/2} \over (1-x)(1-x^2) ... (1-x^m)} due to Euler. The efficiency of the algorithm deteriorates as |x| \uparrow 1, but much more slowly than in previous algorithms. The key lemma is a two-sided bound on the Dedekind eta function at pure imaginary argument, η(iy), that is sharp at the two endpoints y=0,\infty and is accurate to within 9.1% over the entire interval 0 < y < \infty.