\section{Appendix}
\subsection{Averaging over injected momenta}
\label{ss:momentumaveraging}
Suppose a normal distribution of the momentum offset centered about zero.
$$\rho(\delta) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\onehalf\left(\frac{\delta}{\sigma}\right)^2}$$
The average momentum for a particular kicker value is
\begin{eqnarray}
\vev{\delta}
&=&\left(\frac{1}{\int_{\delta_{min}}^{\delta_{max}} \rho(\delta) d\delta}\right)\int_{\delta_{min}}^{\delta_{max}} \delta\rho(\delta) d\delta\nonumber\\
&=&N(\delta_{max},\delta_{min})\int_{\delta_{min}}^{\delta_{max}} \delta  \frac{1}{\sigma\sqrt{2\pi}} e^{-\onehalf\left(\frac{\delta}{\sigma}\right)^2}  d\delta\nonumber\\
&=&N(\delta_{max},\delta_{min}) \frac{1}{\sigma\sqrt{2\pi}}\int_{\delta_{min}}^{\delta_{max}} (-\sigma^2)\der{}{\delta} \left( e^{-\onehalf\left(\frac{\delta}{\sigma}\right)^2}\right)  d\delta\nonumber\\
&=&N(\delta_{max},\delta_{min}) \frac{-\sigma}{\sqrt{2\pi}}\left( e^{-\onehalf\left(\frac{\delta}{\sigma}\right)^2}\right)_{\delta_{min}}^{\delta_{max}}\nonumber\\
&=&N(\delta_{max},\delta_{min}) \frac{-\sigma}{\sqrt{2\pi}}\left( e^{-\onehalf\left(\frac{\delta_{max}}{\sigma}\right)^2}-e^{-\onehalf\left(\frac{\delta_{min}}{\sigma}\right)^2}\right)\label{eq:algebramomentumaverage}
\end{eqnarray}
The normalization
\def\erf{{\rm erf}}
\begin{eqnarray*}
\frac{1}{N(\delta_{max},\delta_{min})}&=& \int_{\delta_{min}}^{\delta_{max}} \frac{1}{\sigma\sqrt{2\pi}} e^{-\onehalf\left(\frac{\delta}{\sigma}\right)^2}  d\delta\\
&=& \int_{\frac{\delta_{min}}{\sqrt{2}\sigma}}^{\frac{\delta_{max}}{\sqrt{2}\sigma}} \frac{1}{\sigma\sqrt{2\pi}} e^{-t^2}  \sqrt{2}\sigma d t\\
&=&\frac{1}{\sqrt{\pi}}\frac{\sqrt{\pi}}{2}\left(\erf(\delta_{max}/(\sqrt{2}\sigma))-\erf(\delta_{min}/(\sqrt{2}\sigma)\right)
\end{eqnarray*}
The variance depends on 
\begin{eqnarray*}
\vev{\delta^2}&=& N(\delta_{max},\delta_{min})\int_{\delta_{min}}^{\delta_{max}} \delta^2  \frac{1}{\sigma\sqrt{2\pi}} e^{-\onehalf\left(\frac{\delta}{\sigma}\right)^2}  d\delta\\
&=& N \frac{1}{\sigma\sqrt{2\pi}}\int_{\delta_{min}}^{\delta_{max}} \delta^2  e^{-\onehalf\left(\frac{\delta}{\sigma}\right)^2}  d\delta\\
&=& N \frac{1}{\sigma\sqrt{2\pi}}\int_{\delta_{min}}^{\delta_{max}}
\sigma^2\left(-\der{}{\delta} \left(\delta e^{-\onehalf\left(\frac{\delta}{\sigma}\right)^2}\right)  d\delta+ e^{-\onehalf\left(\frac{\delta}{\sigma}\right)^2}  d\delta\right)\\
&=& N \frac{1}{\sigma\sqrt{2\pi}}
\left(-\sigma^2\delta e^{-\onehalf\left(\frac{\delta}{\sigma}\right)^2}\right)_{\delta_{min}}^{\delta_{max}}+\sigma^2\\
&=& N \frac{\sigma}{\sqrt{2\pi}}
\left(-\delta e^{-\onehalf\left(\frac{\delta}{\sigma}\right)^2}\right)_{\delta_{min}}^{\delta_{max}}+\sigma^2\\
\end{eqnarray*}
\subsection{Averaging over injected angular distribution $x_{inf}^\prime$} 
\label{ss:angulardistribution}
The injected angular distribution is approximately Gaussian (Figure~\ref{fig:offsetangle}). 
$$\rho(x^\prime) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\onehalf(\frac{\xp}{\sigma})^2}$$
The width and centroid of the momentum slice captured for a particular kick depends quadratically on $\xpinf$.
Averaging over the angular distribution of the injected beam, we replace $\xpinf^2$ in Equations~\ref{eq:widthcaptured} and~\ref{eq:averagecaptured}
with
\begin{eqnarray}
\vev{\xpinf^2}&=&\left(\frac{1}{\int_{\xp_{min}}^{\xp_{max}}\rho(\xp)d\xp}\right)\int_{\xp_{min}}^{\xp_{max}}\rho(\xp) {\xp}^2 d\xp\nonumber\\
&=&N(\xp_{min},\xp_{max})\frac{1}{\sigma\sqrt{2\pi}}\int_{\xp_{min}}^{\xp_{max}}\sigma^2\left(-\der{}{\xp}(\xp e^{-\onehalf(\frac{\xp}{\sigma})^2})d\xp +e^{-\onehalf(\frac{xp}{\sigma})^2}d\xp\right)\nonumber\\
&=&-N(\xp_{min},\xp_{max})\frac{\sigma}{\sqrt{2\pi}}\left(\xp e^{-\onehalf(\frac{\xp}{\sigma})^2}\right)_{\xp_{min}}^{\xp_{max}}+\sigma^2\label{eq:avexp2}
\end{eqnarray}
where
\begin{eqnarray*}
\frac{1}{N(\xp_{min},\xp_{max})}&=&\int_{\xp_{min}}^{\xp_{max}}\frac{1}{\sigma\sqrt{2\pi}}e^{-\onehalf(\frac{\xp}{\sigma})^2}d\xp\\
&=&\frac{1}{2}\left(\erf(\xp_{max}/(\sqrt{2}\sigma))-\erf(\xp_{min}/(\sqrt{2}\sigma))\right)
\end{eqnarray*}
and
\begin{eqnarray}
\xp_{min}&=& -[A^2-(\xinf-\theta_k\beta)^2]^{1/2}\nonumber\\
\xp_{max}&=& -\xp_{min}\label{eq:xpmax}
\end{eqnarray}
Substitution of ~\ref{eq:xpmax} into  ~\ref{eq:avexp2} tells us that
\begin{eqnarray*}
\vev{\xpinf^2}&=& -N(\xp_{min},\xp_{max})\frac{\sigma}{\sqrt{2\pi}}2 \xp_{max}e^{-\onehalf(\frac{\xp_{max}}{\sigma})^2} + \sigma^2
\end{eqnarray*}
What if there is a angluar offset in addition to the spread. Then
$$\rho(\xp) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\onehalf(\frac{\xp-\xp_0}{\sigma})^2}$$
If $y=\xp-\xp_0$ then $dy=d\xp$, $y_{min}=\xp_{min} -\xp_0$  and $y_{max}=\xp_{max}-\xp_0$.
\begin{eqnarray*}
 N^\prime(\xp_{min},\xp_{max})\int_{\xp_{min}}^{\xp_{max}} f(\xp) \frac{1}{\sigma\sqrt{2\pi}}e^{-\onehalf(\frac{\xp-\xp_0}{\sigma})^2}d\xp 
&=& N^\prime\frac{1}{\sigma\sqrt{2\pi}}\int_{y_{min}}^{y_{max}}  f(y+\xp_0) e^{-\onehalf(\frac{y}{\sigma})^2} dy\\
\end{eqnarray*}
If $f(\xp)={\xp}^2$ then
\begin{eqnarray*}
 N^\prime\int_{\xp_{min}}^{\xp_{max}} {\xp}^2 \frac{1}{\sigma\sqrt{2\pi}}e^{-\onehalf(\frac{\xp-\xp_0}{\sigma})^2}d\xp 
&=& N^\prime\frac{1}{\sigma\sqrt{2\pi}}\int_{y_{min}}^{y_{max}} (y^2+2y\xp_0+(\xp_0)^2) e^{-\onehalf(\frac{y}{\sigma})^2} dy\\
&=& N^\prime\frac{\sigma}{\sqrt{2\pi}}\left(-y e^{-\onehalf(\frac{y}{\sigma})^2}\right)_{y_{min}}^{y_{max}} +\sigma^2 -2N^\prime \frac{\sigma}{\sqrt{2\pi}}\xp_0\left( e^{-\onehalf(\frac{y}{\sigma})^2}\right)_{y_{min}}^{y_{max}}  +(\xp_0)^2\\
\end{eqnarray*}
where
\begin{eqnarray*}
\frac{1}{N^\prime(\xp_{min},\xp_{max})}&=&\int_{y_{min}}^{y_{max}}\frac{1}{\sigma\sqrt{2\pi}}e^{-\onehalf(\frac{y}{\sigma})^2}d\xp\\
&=&\frac{1}{2}\left(\erf(y_{max}/(\sqrt{2}\sigma))-\erf(y_{min}/(\sqrt{2}\sigma))\right)
\end{eqnarray*}

\subsection{Capture effiency}
Equation~\ref{eq:algebramomentumaverage} gives the average momentum in each kicker slice. Equation~\ref{eq:avexp2} gives $\vev{{\xp}^2}$ in each slice.
Assuming no correlation between momentum, injection angle and injection time,
the fraction of the injected beam captured in that kicker slice is
\begin{eqnarray*}
\epsilon&=& \int_{\delta_{min}}^{\delta_{max}}\rho_{\delta}(\delta)d\delta\int_{\xp_{min}}^{\xp_{max}}\rho_\xp(\xp)d\xp \int_{t_i}^{t_{i+1}}\rho_{pulse}(t)dt
\end{eqnarray*} 
