Added real overlapping integrated celerite timescale expression to appendix
@@ -620,17 +620,31 @@ This is a figure.}
\section{Exposure time integration for celerite models}
+In this section we give expressions for the {\it exposure-time}
+integrated versions of the \celerite\ kernel and its power
+spectrum. It turns out that the time-integrated \celerite\ kernel
+remains expressible as a semi-separable matrix when the exposures
+do not overlap, but when exposures overlap (or for zero time
+delay), the kernel no longer is semi-separable.
+
\paragraph{Real term}
For a real term:
\begin{equation}
-k(\tau) = a\,\exp\left(-c\,(t_i - t_j)\right)
+%k(\tau) = a\,\exp\left(-c\,(t_i - t_j)\right)
+k(\tau) = a\,\exp\left(-c\,\tau\right)
\end{equation}
-the integrated kernel is
+where $\tau = \vert t_i-t_j \vert$.
+The integrated kernel is
\begin{proof}{celerite-integral}
-k_\Delta(\tau) = \frac{2}{\left(c\,\Delta\right)^2}\,\left[
- \cosh(c\,\Delta) - 1
-\right]\,k(\tau)
+k_\Delta(\tau) = \frac{2}{\left(c\,\Delta\right)^2}\,k(\tau)\,
+ \begin{cases}
+ \cosh(c\,\Delta) - 1 & \tau > \Delta \\
+ c(\Delta - \tau) e^{c\tau} + \cosh{(c\tau)} e^{-c(\Delta - \tau)} -1 & \tau \le \Delta
+ \end{cases}
+%k_\Delta(\tau) = \frac{2}{\left(c\,\Delta\right)^2}\,\left[
+% \cosh(c\,\Delta) - 1
+%\right]\,k(\tau)
\end{proof}
where $\Delta$ is the integration time.
