|^2 .$$ If we expand the pseudo-multipoles in terms of the true multipoles using equation (\[eq:5-25\]), we find $$\begin{aligned}
{\rmn{cov}} (\tilde{C}_l^E,\tilde{C}_{l'}^E )
&=& \frac{2}{(2l+1)(2l'+1)} \sum_{mm'} \left|\sum_{LM}
{}_{+}I_{(lm)(LM)}{}_{+}I^*_{(lm)'(LM)}C_L^E + {}_{-}I_{(lm)(LM)}
{}_{-}I^*_{(lm)'(LM)}C_L^B \right|^2,
\label{eq:5-37}\\
{\rmn{cov}} (\tilde{C}_l^B,\tilde{C}_{l'}^B ) 
&=& \frac{2}{(2l+1)(2l'+1)} \sum_{mm'} \left|\sum_{LM}
{}_{+}I_{(lm)(LM)} {}_{+}I^*_{(lm)'(LM)}C_L^B + {}_{-}I_{(lm)(LM)}
{}_{-}I^*_{(lm)'(LM)}C_L^E \right|^2, \label{eq:5-38} \\
{\rmn{cov}} (\tilde{C}_l^E,\tilde{C}_{l'}^B ) 
&=& \frac{2}{(2l+1)(2l'+1)} \sum_{mm'} \left|\sum_{LM}
{}_{+}I_{(lm)(LM)} {}_{-}I^*_{(lm)'(LM)}C_L^E + {}_{-}I_{(lm)(LM)}
{}_{+}I^*_{(lm)'(LM)}C_L^B
 \right|^2.
\label{eq:5-39}\end{aligned}$$ Evaluating these expressions numerically is straightforward in principle, but is computationally expensive, with an operation count of $O(l^6)$ if multipoles up to $l$ are retained. This is prohibitive for $l$ above a few hundred, and so, in general, there is a need for faster approximate calculations. One such approximation is to compute the covariances from Monte-Carlo simulations; this has the benefit of allowing one easily to include a number of real-world effects that are more difficult to include in the analytic calculation. An alternative, which is developed here in Section \[sec:approx-cov\], is to use analytic approximations (that can be calibrated with simulations if required). As we show later, it is difficult to find good analytic approximations at low $l$ because of the expected sharp rise in the polarization $C_l$s there, and the increased importance of $E$-$B$ mixing. A similar 