Nome

Remarks

For real $k \in (0, 1)$ the nome is real with $0 < q < 1$, and it increases from $0$ to $1$ as $k$ goes from $0$ to $1$. Example: at $k = 1/\sqrt2$ one has $k = \sqrt{1 - k^2}$, so $K = K'$ and $q = e^{-\pi} \approx 0.0432$. As $k \to 0$, $K \to \pi/2$ and $K' = \log(4/k) + o(1)$, so $q = k^2/16 + O(k^4)$. Non-example: $\tau = 0$ or real $\tau$ gives $\abs{q} = 1$, which is not a nome, since the theta series in $q$ do not converge there. Notation varies: some authors, especially in the theory of modular forms, use $q = e^{2\pi i\tau}$ instead and call it the nome as well. With the nome $q = e^{\pi i\tau}$ defined above, the Jacobi theta function is $\vartheta(z, \tau) = \sum_{n \in \Z} q^{n^2} e^{2\pi i n z}$.