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By Per-Olov Lowdin


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This yields the expression gp(X, y, Z) = CK~pXKyAZp, I , 1. II K +i +p =p, (1 18) with coefficients The polynomial of Eq. (1 18) is a homogeneous function of degree p and could have been given without relying on other formulas, since Eq. ( I 19) follows from Eq. ( I 18). However, the derivation given here may illustrate how the most general form of a homogeneous function, which is given by Eq. (1 18), follows directly from the definition stated by Eq. (1 1 1). In the three-dimensional case the identity of Eq.

Where the two independent solutions of Eq. ,the harmonics are centered on this origin. A convenient choice for specifying the location of P are spherical coordinates r, 8, cp [see Eq. (10) and Fig. 21. However, the argument of each harmonic may also be denoted by the vector r which connects the origin 0 with the field point P,directing from 0 to P, provided that the reference system is given. Then, this shorthand notation leaves no doubt about its meaning. It has been shown that there are at most (21 1) linearly independent regular solid spherical harmonics of degree I which are sufficient to construct any general regular solid spherical harmonic according to Eq.

12b). Then, the distance of point P from center B is smaller than the displacement of the function from A to B. a. Distance of P greater than displacement. If c/r' < 1, Taylor's expansion of l/r is given by Rotation and Translation of Solid Spherical Harmonics 53 see Eq. (218) and Figs. 9 and 12a. Consideringthat x = x', y = y', z = z' + c, with respect to either the the application of the operator Q,,+(d/dz)'-"' unprimed or the primed coordinates yields the relation or by virtue of Eqs. (229) and (230) Since r' = r> and c = r< , it is expedient to write this for later purposes as if 8, denotes the angle between r> and the z axis.

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