$\def\ai{\operatorname{ai}} \def\bi{\operatorname{bi}} \def\Ai{\operatorname{Ai}} \def\Bi{\operatorname{Bi}} $ Airy Ai Zero $\ai_n$ gives the $n$th zero of the Airy Ai function and Airy Bi Zero $\bi_n$ of Airy Bi. The analytic continuation of $\ai_n,\bi_n$ use Bessel J Zero $j_{v,x}$, and Bessel Y Zero $y_{v,x}$
$$-\left(\frac32j_{\frac13,n-\frac16}\right)^\frac23=\ai_n,n\in\Bbb N$$ 
shown here and $$-\left(\frac32 y_{\frac13,n-\frac16}\right)^\frac23=\bi_n,n\in\Bbb N$$ 
shown here.
Using our extensions, we find that “$\ai_{x-\frac12}=\bi_x$“, so we ignore $\bi_n$‘s inverse for now. Now invert the extensions using DLMF 9.9ii and the Airy Phase function $\theta(x)$:
$$-\left(\frac32j_{\frac13,\frac{\theta(x)}\pi+n-\frac16}\right)^\frac23=x;\bi_n<x<\bi_{n+1}, n\in\Bbb N, $$
Unfortunately, $\frac{\theta(x)}\pi=\frac{\tan^{-1}\left(\frac{\Ai(x)}{\Bi(x)}\right)}\pi$ has discontinuities at the aforementioned $\bi_n$:
$\tag1$
There cannot be a left inverse of the discrete $\ai_n,n\in\Bbb N$ and approximations for it are not the right inverse. On the other hand, if one superimposes $\frac{\theta(x)}\pi+n,n\in\Bbb N$, then we want the highlighted branch which will give the right and most accurate left inverse for $\ai_n$.
$\tag2$
The piecewise function for the highlighted graph is:
$$\cases{\frac{\theta(x)}\pi,x>\bi_1\\ \frac{\theta(x)}\pi+1, \bi_1<x<\bi_2\\\frac{\theta(x)}\pi+2, \bi_2<x<\bi_3\\\rule{13.65mm}{0ex} \vdots\\\frac{\theta(x)}\pi+n, \bi_n<x<\bi_n}$$
but it is unwieldy. What is an equation for $(2)$’s highlighted graph given $(1)$? It could be a meromorphic extension.
The DLMF Airy Modulus function $\text M(x)=\sqrt{\Ai^2(x)+\Bi^2(x)}$ appears after differentiating:
$$\begin{align}\frac d{dx}\tan^{-1}\left(\frac{\Ai(x)}{\Bi(x)}\right)=-\frac1{\pi\left(\Ai^2(x)+\Bi^2(x)\right)}\implies \frac{\theta(x)}\pi\text{ without discontinuities}=-\frac1\pi\int\frac{dx}{\Ai^2(x)+\Bi^2(x)}=-\frac1\pi\int \text M^{-2}(x)dx\end{align}$$
The graph of the derivative has no discontinuities, but how does one integrate it so there are none afterwards?
