# How to compute Weber polynomials efficiently?

Given $\tau\in H$ (up-half plane) and $q=e^{2\pi i \tau}$, Weber polynomail is defined as

$$f(\tau)=q^{-\frac{1}{48}}\prod_{i=0}^{\infty}(1+q^{i-\frac{1}{2}}).$$

My question is: How can I compute a product of unlimited sequence? Anyway, I have to finish $f(\tau)$ in finite steps. Of course, the faster the better.

• $f(\tau)$ is not a Weber polynomial, but a Weber function. – Alex M. May 4 '18 at 13:15

• Thanks a lot! I still have some small questions above your paper: (1) To compute $\Delta(\tau)$ on page 339, do you mean to approximate $\sum_{n\geq 1}...$ using Taylor method? (2) Does the defintion of $F(z)$ on page 340 say that $F(z)=1+\sum_{n\geq 1}...$ by cancelling the $z$ and the exponents $24$ and $1/24$? (3) What are $G$ and $I$ mentioned on page 341? – Licheng Wang May 5 '18 at 2:07