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But so we have simply

this is the first term in the asymptotic expansion, where is an arbitrary constant,
valid away from the boundary layers (see §2.6). (If we were to apply the boundary
condition on x = 1/2, then we would deduce that which turns out to be
correct, as we shall see below.)
For the boundary layer near x = 1/2, we write and
which gives (from (5.94))

But the dominant contribution to the integral will come from the behaviour of N
outside the boundary layers i.e. we use When we do this, the
integral term yields the result

and this is used in (5.96), together with the boundary condition to
give

The corresponding solution in the boundary layer at the other end is obtained from this
result by forming where Note that, because the boundary
condition has been used here, is now determined (in a way analogous to matching)
and so away from the boundary layers. This concludes all that we will
write about this very different type of boundary-layer problem; see Pao & Tchao
(1970) for more details.

5.4 SEMI- AND SUPERCONDUCTORS
The study of semiconductors and of superconductors, as it has unfolded over the last
50 years or so, has thrown up any number of interesting and important equations that
describe their properties and design characteristics. We will look at three fairly typical
examples: E5.16 Josephson junction; E5.17 A p-n junction; E5.18 Impurities in a
semiconductor.