Nonconstant-
Limit
Suppose that
. In this limit, Equation (5.78) reduces to
![$\displaystyle \frac{d}{dp}\!\left(p^2\,\frac{dY_e}{dp}\right)-[-{\rm i}\,(Q-Q_E-Q_e)]\,\frac{B(p)}{C(p)}\,p^2\,Y_e=0.$](img2227.png) |
(5.99) |
In the various nonconstant-
regimes considered in Section 5.11, the previous equation takes the form
![$\displaystyle \frac{d}{dp}\!\left(p^2\,\frac{dY_e}{dp}\right)-G\,p^{k+2}\,Y_e = 0,$](img2228.png) |
(5.100) |
where
is real and non-negative, and
is a complex constant. Let
. The previous equation
yields
![$\displaystyle \frac{d^2 U}{dp^2} - G\,p^k\,U = 0.$](img2230.png) |
(5.101) |
This equation is identical in form to Equation (5.88), which we have already solved. Indeed, the solution that
is bounded as
has the small-
expansion (5.90), where
,
, and
. Matching to Equation (5.83) yields
![$\displaystyle \skew{6}\hat{\mit\Delta} = -\frac{\pi\,\nu^{1-2\nu}{\mit\Gamma}(\nu)}{{\mit\Gamma}(1-\nu)}\,\,G^{-\nu}.$](img2231.png) |
(5.102) |
The layer width in
-space again scales as
. This width must be
less that
. As before, the neglect of the term involving
in Equation (5.74) is
justified provided that
.