Fourier Transformation
Equations (5.57)–(5.60) are most conveniently solved in Fourier transform space [7,15,18].
Let

(5.72) 
et cetera. The Fourier transformed layer equations become
where, for a tearing parity solution [10],

(5.77) 
as
. Here,
is an arbitrary constant, and use has been made of Equation (5.71).
In accordance with our previous discussion, we have neglected the parallel transport terms in Equation (5.74).
Let us also ignore the term
. This approximation can
be justified a posteriori. It is equivalent to the neglect of the contribution of ion parallel dynamics to the linear plasma
response in the resonant layer, and effectively decouples Equation (5.76) from Equations (5.73)–(5.75)
[22]. Equations (5.73)–(5.75) reduce to [7,15]

(5.78) 
where
and
is the normalized, Fourier transformed, electron fluid streamfunction. It is easily
demonstrated that

(5.82) 
as
.
Hence, the boundary conditions on Equation (5.78) are that is bounded as
, and

(5.83) 
as
, where is an arbitrary constant. Here, use has been made of Equation (5.77). In the following, we shall assume that
,
, and
, for the
sake of simplicity.