After this question of mine related to the problem of approximate solutions of $$\large\color{red}{\operatorname{\vartheta}_3}\left(0,x\right)=k$$ when $k$ is large, continuing the previous work (just for the art for art's sake) thanks to @uranix's answer and suggestions, I ended with tha approximate asymptotic inverse $$\large x=\exp\left(-A^2\frac{\pi }{k^2} \right)\qquad \text{where} \qquad A=\color{red}{\vartheta _3}\left(0,e^{-\pi k^2}\right)$$ which is quite good as soon as $k \geq 1.5$ (for this value the error is $6.20\times 10^{-5}$).
Concerning the norm $$\Phi=\int_{\frac 32}^\infty \big[{\vartheta}_3\left(0,x\right)-k\big]^2\,dk=1.11\times 10^{-10}$$
What is surprising (at least to me) is to see the inverse function involving the function itself.
Do we know similar cases ?
