deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 21fce2e..d6f924a 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
Therefore we can expand out the phase evolution to be  \begin{equation}  \phi(t) \approx 2\pi \sum_{l=0} \frac{f^{(l)}}{(l+1)!} \left(t(1+\dot{\delta}_k \left(t\left(1+\dot{\delta}_k  + \frac{\ddot{\delta}_k}{2}t)-t_0+\delta_k+t_k\right)^{(l+1)}. \frac{\ddot{\delta}_k}{2}t\right)-t_0+\delta_k+t_k\right)^{(l+1)}.  \end{equation}