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92 2. Introductory applications

Near x = 0, we write and, in order to match, we require y = O(1), so we
write equation (2.94) then gives

and hence we select Thus in the transition layer we have the equation

and we seek a solution where

(The lower limit in the integral here is simply a convenience; with this choice, the
arbitrary constant following the second integration is Note that this integral exists
for These arbitrary constants are determined by matching; from
(2.96a,b) we obtain directly that

From (2.97), we first write

now let us introduce the constant

(where is the gamma function), then

and matching (2.98) and (2.100) requires and The first term
of the asymptotic expansion valid in the transition layer (around x = 0) is therefore