# 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 two-layer approximation is equivalent to the well-known 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 non-negative, 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 .