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146 3. Further applications

for into (3.96) and then cancel everywhere), to give

and then, for this can be written

This is easily integrated to produce the solution

where the term involving the arbitrary constant of integration is suppressed because it
is less singular than the term retained. Thus the asymptotic expansion (3.89) becomes

which exhibits a breakdown where exactly as for (3.87) (see Q3.18)—
just what we most feared! But there is a very important difference here: the original
expansion broke down where and we still required a solution valid on
x = 0; now we have a breakdown at but, because x = 0 corresponds
to

obtained directly from (3.99), we require tobenosmaller than Of course,
the burning question now is: are we allowed to use (3.99) with and
hence define The answer is quite surprising.
We have seen that

where is the constant it is fairly straightforward to show that

where the are constants bounded as Thus the asymptotic expansion (3.89),
for the strained coordinate, becomes