6.7 Appendix A

In this appendix recurrence relations for the coefficients ? ^m _{? 1 ?},( ? ₁), defined for oblate spheroids by eq. (6.108), and further useful formulae for these coefficients will be derived, cf. ref. [42] where a slightly different notation is used.

It may, first of all, be remarked that eq. (6.108) is in particular valid on the negative z-axis ( ? = ?1) with z < 2 a ? ₁ (see fig. 6.18). It is easily seen that on that axis

Figure 6.18: Region of validity of expansion eq. (6.108), within dashed oblate spheroid.

where ? ?( ?, ?) and ?( ?, ?) are the first two oblate spheroidal coordinates of a point in the shifted coordinate frame with origin at (0,0,2 d).

It furthermore follows with eqs. (6.14) and (6.13) that

and

Using eqs. (6.207) and (6.103) one obtains

From eq. (6.5) one has

Since P ^m _?(1) = ? _m0, one finally finds from the last two equations

Substituting eqs. (6.208), (6.209) and (6.212) into eq. (6.108) for ? = ?1, one obtains, using d = a ? ₁, for m = 0 the formula

For m = 1 one has to divide eq. (6.108) by (1 ? ? ²) ^1/2 and then take the limit ? ? ?1. One obtains

With eqs. (6.103) and (6.5) it...