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A table of curly U values as originally defined in Lai (2012) {#InclinationEccentricity_Ummtable}
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m'=-2
-----
\f{eqnarray*}{
    \mathcal{U}_{-2,-2}&=&\sqrt{\frac{3\pi}{10}}\frac{1}{4}(1+cos\theta)^2\\
    \mathcal{U}_{-1,-2}&=&-\sqrt{\frac{3\pi}{10}}\frac{1}{2}
                        \sin\theta(1+\cos\theta)\\
    \mathcal{U}_{0,-2}&=&\sqrt{\frac{\pi}{5}}\frac{3}{4}\sin^2\theta\\
    \mathcal{U}_{1,-2}&=&-\sqrt{\frac{3\pi}{10}}\frac{1}{2}
                        \sin\theta(1-\cos\theta)\\
    \mathcal{U}_{2,-2}&=&\sqrt{\frac{3\pi}{10}}\frac{1}{4}(1-\cos\theta)^2\\
\f}

m'=0
----
\f{eqnarray*}{
    \mathcal{U}_{-2,0}&=&-\sqrt{\frac{6\pi}{5}}\frac{1}{4}\sin^2\theta\\
    \mathcal{U}_{-1,0}&=&-\sqrt{\frac{6\pi}{5}}\frac{1}{2}
                        \sin\theta\cos\theta\\
    \mathcal{U}_{0,0}&=&-\sqrt{\frac{\pi}{5}}\frac{1}{2}(3\cos^2\theta-1)\\
    \mathcal{U}_{1,0}&=&\sqrt{\frac{6\pi}{5}}\frac{1}{2}
                        \sin\theta\cos\theta\\
    \mathcal{U}_{2,0}&=&-\sqrt{\frac{6\pi}{5}}\frac{1}{4}\sin^2\theta
\f}

m'=2
----
\f{eqnarray*}{
    \mathcal{U}_{-2,2}&=&\sqrt{\frac{3\pi}{10}}\frac{1}{4}(1-\cos\theta)^2\\
    \mathcal{U}_{-1,2}&=&\sqrt{\frac{3\pi}{10}}\frac{1}{2}
                        \sin\theta(1-\cos\theta)\\
    \mathcal{U}_{0,2}&=&\sqrt{\frac{\pi}{5}}\frac{3}{4}\sin^2\theta\\
    \mathcal{U}_{1,2}&=&\sqrt{\frac{3\pi}{10}}\frac{1}{2}
                        \sin\theta(1+\cos\theta)\\
    \mathcal{U}_{2,2}&=&\sqrt{\frac{3\pi}{10}}\frac{1}{4}(1+\cos\theta)^2\\
\f}