Spherical Harmonics

5 Spherical Harmonics 7 5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.1 Harmonic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.2.3 Rotational invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.2.4 Fast Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.2.5 Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.3 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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