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74 2. Introductory applications

and then equation (2.46b) is

or

The general expression for is therefore

and so we have the expansion

and this is to be matched with (2.44). From (2.44) we obtain

and, correspondingly from (2.49), we have

or

which matches only if (because the term in must be eliminated).
The solution valid for is therefore

and this expansion is defined on X = 0, yielding

We observe, in this example, that the value of the function on x = 0 is well-defined
from (2.51), but that it diverges as This demonstrates the important property
that we require, for a solution to exist, that the asymptotic representation be defined