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256 5. Some worked examples arising from physical problems

and then from (5.130) we obtain

with

This set is very easily solved to give the asymptotic solution

and it is immediately evident that this expansion breaks down when let us
write and to produce the new equation

Again, we seek a straightforward solution

so that (5.132) gives

and so on, together with the requirement to match to (5.131).
The general solution to equation (5.133a) is

where A is an arbitrary constant; with we obtain

and hence matching to the first term in (5.131) requires the choice A=1. Thus we
have

the next term can be found similarly and leads to