\subsection{Dependence of Pitch Correction on $\omega_v$}
\label{ss:intermsofpitchangle}
\begin{eqnarray}
\omega^\prime&=&\omega_a\left(1-\frac{(\beta_z^0)^2}{2}\left[\frac{\gamma}{\gamma+1} -
\frac{1}{2}\omega_a\beta_{xy}^2\left(\frac{2\omega_v(\frac{\gamma^2}{\gamma+1}\frac{\omega_v}{\omega})
-\omega_a(\frac{\gamma^2}{(\gamma+1)^2}+\gamma^2\frac{\omega_v^2}{\omega^2})}{\omega_v^2-\omega_a^2}\right)\right]\right)\\
&=&\omega_a\left(1-\frac{(\beta_z^0)^2}{2}\left[\frac{\gamma}{\gamma+1} -
\frac{1}{2}\omega_a\beta_{xy}^2\left(\frac{(\omega^2_v-\omega_a^2)(\frac{2}{\omega}\frac{\gamma^2}{\gamma+1}
-\frac{\omega_a}{\omega^2}\gamma^2) + \omega_a^2(\frac{2}{\omega}\frac{\gamma^2}{\gamma+1}-\frac{\omega_a}{\omega^2}\gamma^2)
-\omega_a\frac{\gamma^2}{(\gamma+1)^2}}
{\omega_v^2-\omega_a^2}\right)\right]\right)\\
&=&\omega_a\left(1-\frac{(\beta_z^0)^2}{2}\left[\frac{\gamma}{\gamma+1} -
\frac{1}{2}\frac{\omega_a}{\omega}\beta_{xy}^2\left(((2\frac{\gamma^2}{\gamma+1}
-\frac{\omega_a}{\omega}\gamma^2) +\frac{ \omega_a^2(2\frac{\gamma^2}{\gamma+1}-\frac{\omega_a}{\omega}\gamma^2)
-\omega\omega_a\frac{\gamma^2}{(\gamma+1)^2}}
{\omega_v^2-\omega_a^2}\right)\right]\right)
\end{eqnarray}
Using $\omega_a/\omega = a_\mu\gamma$, $\beta_{xy}^2 = \frac{\gamma^2-1}{\gamma^2}$ and at the magic momentum $a_\mu\gamma = \frac{\gamma}{\gamma^2-1}$, 
\begin{eqnarray}
\omega^\prime&=&\omega_a\left(1-\frac{(\beta_z^0)^2}{2}\left[\frac{\gamma}{\gamma+1} -
\frac{1}{2}\frac{1}{\gamma}\left((2\frac{\gamma^2}{\gamma+1}
-\frac{\gamma^3}{\gamma^2-1}) +\frac{ \omega_a^2(2\frac{\gamma^2}{\gamma+1}-\frac{\gamma^3}{\gamma^2-1})
-\frac{\omega_a^2(\gamma^2-1)}{\gamma}\frac{\gamma^2}{(\gamma+1)^2}}
{\omega_v^2-\omega_a^2}\right)\right]\right)\\
&=&\omega_a\left(1-\frac{(\beta_z^0)^2}{2}\left[\frac{\gamma}{\gamma+1} -
\frac{1}{2}\gamma\left((2\frac{1}{\gamma+1}
-\frac{\gamma}{\gamma^2-1}) +\frac{ \omega_a^2(2\frac{1}{\gamma+1}-\frac{\gamma}{\gamma^2-1}
-\frac{(\gamma^2-1)}{\gamma}\frac{1}{(\gamma+1)^2})}
{\omega_v^2-\omega_a^2}\right)\right]\right)\\
&=&\omega_a\left(1-\frac{(\beta_z^0)^2}{2}\left[\frac{\gamma}{\gamma+1} -
\frac{1}{2}\gamma\left((\frac{2\gamma-2 -\gamma}{\gamma^2-1}) +\frac{ \omega_a^2(\frac{2\gamma-2 -\gamma}{\gamma^2-1})
-\frac{\gamma-1}{\gamma(\gamma+1)})}
{\omega_v^2-\omega_a^2}\right)\right]\right)\\
&=&\omega_a\left(1-\frac{(\beta_z^0)^2}{2}\left[\frac{\gamma}{\gamma+1} -
\frac{1}{2}\gamma\left(\frac{\gamma-2}{\gamma^2-1} +(\frac{ \omega_a^2(\frac{(\gamma-2)}{\gamma^2-1}
-\frac{\gamma-1}{\gamma(\gamma+1)})}
{\omega_v^2-\omega_a^2}\right)\right]\right)\\
&=&\omega_a\left(1-\frac{(\beta_z^0)^2}{2}\left[\frac{\gamma(\gamma-1) -
\frac{\gamma}{2}(\gamma-2)}{\gamma^2-1}-\frac{\gamma}{2}\frac{ \omega_a^2(\frac{(\gamma-2)}{\gamma^2-1}
-\frac{\gamma-1}{\gamma(\gamma+1)})}
{\omega_v^2-\omega_a^2}\right]\right)\\
&=&\omega_a\left(1-\frac{(\beta_z^0)^2}{2}\left[\frac{\gamma^2}{2(\gamma^2-1)}
-\frac{\gamma}{2}\frac{ \omega_a^2\frac{\gamma(\gamma-2) -(\gamma-1)^2}{\gamma(\gamma^2-1)}}
{\omega_v^2-\omega_a^2}\right]\right)\\
&=&\omega_a\left(1-\frac{(\beta_z^0)^2}{2}\left[\frac{\gamma^2}{2(\gamma^2-1)}
-\frac{\gamma}{2}\frac{ \omega_a^2\frac{-1}{\gamma(\gamma^2-1)}}
{\omega_v^2-\omega_a^2}\right]\right)\\
&=&\omega_a\left(1-\frac{(\beta_z^0)^2}{4}\left[\frac{\gamma^2}{\gamma^2-1}
+\frac{ \omega_a^2\frac{1}{(\gamma^2-1)}}
{\omega_v^2-\omega_a^2}\right]\right)
\end{eqnarray}
In terms of the pitch angle $\psi_0 \approx \frac{\beta_z^0}{\beta_{xy}}$
\begin{eqnarray}
\omega^\prime &=& \omega_a\left(1-\frac{\psi_0^2}{4}\left[1+\frac{\omega_a^2}{\gamma^2(\omega_v^2-\omega_a^2)}\right]\right)\label{eq:psiofomegav}
\end{eqnarray}
%&\equiv&\omega_a\left(1-\frac{(\beta_z^0)^2}{4}\left[1+a_\mu
%+\frac{ \omega_a^2\frac{1}{(\gamma^2-1)}}
%{\omega_v^2-\omega_a^2}\right]\right)\\
%&\equiv&\omega_a\left(1-\frac{(\beta_z^0)^2}{4}\left[1+a_\mu
%+\frac{ \omega_a^2}
%{\gamma^2(\omega_v^2-\omega_a^2)}(1+a_\mu)\right]\right)
%\end{eqnarray}

In the limit where $\omega_v\rightarrow 0$,
\begin{eqnarray*}
\omega^\prime &=&\omega_a\left(1-\frac{(\beta_z^0)^2}{4}\right)
\end{eqnarray*}
