\section{Summary}
\label{sec:summary}

\noindent
{\bf Fixed pitch}, with velocity $\beta_z$ parallel to the magnetic field $\bfB=B_z\bfhz$.
\begin{equation}
\omega^\prime =\omega_a(1-\beta_z^2)^{1/2}
\end{equation}

\noindent
{\bf Time dependent pitch.} Vertical betatron motion with amplitude $\beta_z^0$. ($\beta_z^0\sim\psi$ where $\psi$ is the pitch angle) for arbitrary (but non zero) vertical betatron frequency $\omega_v$ and at the magic momentum. To order $(\beta_z^0)^2$
\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)
\end{eqnarray*}
{\bf Time dependent pitch where pitching frequency is much greater than $\omega_a$}. Vertical betatron motion with amplitude $\beta_z^0$ in the $\omega_v\gg\omega_a$ limit, at the magic momentum
\begin{eqnarray*}
\omega^\prime&=&\omega_a\left(1- \frac{(\beta_z^0)^2}{4}\left[\frac{\gamma^2}{(\gamma^2-1)}\right]\right)
\end{eqnarray*}

\noindent
{\bf Radial magnetic field}
\begin{equation}
\omega^\prime = \omega_a\left[1 + \left(\frac{B_r}{B_z}\right)^2\left(\frac{1+1/a_\mu}{\gamma^2}\right)^2\right]^{1/2}
\end{equation}

\noindent
{\bf Uniform longitudinal magnetic field}
\begin{equation}
\omega^\prime=\omega_a\left[1 + \left(\frac{B_l}{B_z}\right)^2\left(\frac{1}{a_\mu\gamma}+\frac{1}{\gamma}\right)^2\right]^{1/2}
\end{equation}

\noindent
{\bf Nonuniform longitudinal field in terms of fourier harmonics} to order $(b_n/B_z)^2$ for magnetic field harmonics and $b_n$ defined by
$$\bfB_n=b_n\cos n\theta(\cos\theta\ \bfhx - \sin\theta\ \bfhy)+B_z\bfhz$$ 
\begin{eqnarray*}
\omega^\prime
&=&  \omega_a\left( 1- \frac{b_n^2}{B_z^2}\frac{\gamma^2}{4}\frac{\omega_a^2}{(n\omega)^2-\omega_a^2}\right)\\
\frac{\Delta\omega^\prime}{\omega_a}&=& -\frac{1}{4}\frac{b_n^2}{B_z^2}\gamma^2\frac{\omega_a}{(n\omega)^2-\omega_a^2}
\end{eqnarray*}

\noindent
{\bf Closed orbit electric field contribution}
\begin{equation}
\omega^\prime = \frac{e}{mc}\left(B_z a_\mu -\left(a_\mu-\frac{1}{\gamma^2-1}\right)\beta E_r\right)
\end{equation}
