Constant Limit
Let us suppose that there are two layers in space. In the small layer, suppose that Equation (5.78) reduces to

(5.84) 
when
. Integrating directly, we find that

(5.85) 
for
, where use has been made of Equation (5.83). The twolayer approximation is
equivalent to the wellknown constant approximation [17].
In the large layer, for
, we obtain

(5.86) 
with bounded as
. Asymptotic matching to the small layer solution (5.85) yields the boundary
condition

(5.87) 
as
.
In the various constant linear response regimes considered in Section 5.9, Equation (5.86) reduces to an
equation of the form

(5.88) 
where is real and nonnegative, and is a complex constant. Let
and
, where
. The previous equation transforms into a modified Bessel equation of general order,

(5.89) 
where
. The solution that is bounded as
has the small
expansion [1]

(5.90) 
where
is a gamma function.
A comparison of this expression with Equation (5.87) reveals that

(5.91) 
Note, finally, that
, where denotes the width of the large layer in space. This width must be larger than (i.e., the width of the small layer) in order for the constant approximation to hold. Finally, it is easily demonstrated that the neglect of the term involving in Equation (5.74) is
justified provided that
.