What is the limit $\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$

Question

If $\vartheta_{2}(q)$ is jacobi's theta function, what is the limit $$\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$$ for the nome $q$. I would like to know whether the limit exists or not. If it does, please let me know and provide it's evaluation

Answer 1

We know that $$\vartheta_{2}^{2}(q) = \frac{\vartheta_{2}^{2}(q)}{\vartheta_{3}^{2}(q)}\cdot \vartheta_{3}^{2}(q) = \frac{2kK}{\pi}$$ and $q = e^{-\pi K'/K}$. If $q \to 1$ so that $K'/K \to 0$ which means that $k \to 1$. So we have \begin{align} L &= \lim_{q \to 1}\vartheta_{2}^{2}(q)(1 - q)\notag\\ &= \lim_{k \to 1}\frac{2kK}{\pi}\left(1 - e^{-\pi K'/K}\right)\notag\\ &= \lim_{k \to 1}\frac{2kK}{\pi}\cdot\frac{\pi K'}{K}\cdot\frac{1 - e^{-\pi K'/K}}{\pi K'/K}\notag\\ &= \lim_{k \to 1}2kK'\cdot 1\notag\\ &= 2\lim_{k \to 1}K'\notag\\ &= 2\cdot\frac{\pi}{2} = \pi\notag \end{align} I have used only real variable theory and the limit is for $q \to 1^{-}$. Note that $K(k)$ is a strictly increasing function of $K$ and maps interval $[0, 1)$ to $[\pi/2, \infty)$. And $K'(k)$ is a strictly decreasing function of $k$ which maps $(0, 1]$ to $[\pi/2, \infty)$. Therefore $K'/K$ is a strictly decreasing function which maps $(0, 1)$ to $(0, \infty)$. These basic facts about monotonic nature of $K, K'$ are proved in my blog post.