I have the following ratios I want to compute.

$$ \frac{ \left( \frac{\partial \vartheta_3(v, q)}{\partial v} \right)^2 }{C + \left(\vartheta_3(v, q)\right)^2 }, $$ where $C$ is a constant.

$$ \frac{ \left( \frac{\partial \vartheta_3(v, q)}{\partial q} \right)^2 }{C + \left(\vartheta_3(v, q)\right)^2 }, $$ and $$ \frac{ \left( \frac{\partial \vartheta_3(v, q)}{\partial v} \frac{\partial \vartheta_3(v, q)}{\partial q} \right) }{C + \left(\vartheta_3(v, q)\right)^2 }. $$

Are there closed form or any functional solutions to these?

Here,

$$ \vartheta_3(v, q) = 1 + 2 \sum_{n = 1}^{\infty} q^{n^2} \cos(2nv) $$