To zeroth order in , Equation (11.72) yields

(11.100) 
Solving this equation subject to the boundary condition (11.83), we obtain

(11.101) 
Thus, we again conclude that, to lowest order in our expansion, the magnetic fluxsurfaces in the island region have the constant
structure pictured in Figure 5.7. The island Opoints correspond to
and
(where is an integer), the
Xpoints correspond to
and
, and the magnetic separatrix corresponds to
.
To zeroth order in , Equation (11.71) yields

(11.102) 
where use has been made of Equation (11.101). Given that is an odd function of , it
follows that

(11.103) 
By symmetry,
inside the magnetic separatrix of the island chain.
Let

(11.104) 
Note that
. Equations (11.84) and (11.101) imply that

(11.105) 
To zeroth order in , Equation (11.68) yields

(11.106) 
where use has been made of Equations (11.101) and (11.103). Given that
is an odd function of , it follows that

(11.107) 
By symmetry,
inside the magnetic separatrix of the island chain. Let

(11.108) 
Note that
. Equations (11.85) and (11.101) imply that

(11.109) 
To zeroth order in , Equation (11.69) yields

(11.110) 
where use has been made of Equations (11.101), (11.103), and (11.107). Given that
is an even function of , we can write

(11.111) 
Finally, to zeroth order in , Equation (11.70) gives
where

(11.113) 
and use has been made of Equations (11.101), (11.103), (11.104), (11.107), and (11.108). Moreover,
. Note that
.
Let us write

(11.114) 
where
has the symmetry of , whereas
has the symmetry of
.
It follows from Equations (11.112) and (11.114) that

(11.115) 
which implies that

(11.116) 
where the
operator is defined in Section 8.6, and
is an undetermined fluxsurface function.
Equations (11.112) and (11.114) also yield

(11.117) 
The fluxsurface average (see Section 8.6) of the previous equation gives

(11.118) 
Inside the separatrix of the magnetic island chain, recalling that
, the previous equation reduces to

(11.119) 
whereas outside the separatrix, it gives

(11.120) 
where use has been made of the easily proved identity
.
Thus, we conclude that

(11.121) 
inside the separatrix, and

(11.122) 
outside the separatrix. Here, use has been made of
outside the separatrix.
However, the boundary condition (11.86), combined with Equations (11.101), (11.105), (11.109), and (11.113), implies that
. Finally, Equations (11.117), (11.121), and (11.122) yield

(11.123) 