deletions | additions       diff --git a/Expanding_out_alpha_we_get__.tex b/Expanding_out_alpha_we_get__.tex  index 9d33c22..77fa524 100644  --- a/Expanding_out_alpha_we_get__.tex  +++ b/Expanding_out_alpha_we_get__.tex 
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\end{equation}  where  \begin{align}  z_1 &= \left[\left(\frac{\pi}{3}\right)^{4/3}\dot{f_k}\ddot{f_k}^{1/3}-\frac{\Delta t}{2}\right]\left(\frac{\pi}{3}\right \ddot{f_k})^{1/3}, t}{2}\right]\left(\frac{\pi}{3}\ddot{f_k}\right)^{1/3},  \nonumber \\ z_2 &= \left[\left(\frac{\pi}{3}\right)^{4/3}\dot{f_k}\ddot{f_k}^{1/3}+\frac{\Delta t}{2}\right]\left(\frac{\pi}{3}\right\ddot{f_k})^{1/3}. t}{2}\right]\left(\frac{\pi}{3}\ddot{f_k}\right)^{1/3}.  \end{align}