Evaluation of Integrals
Figure: 11.1
The integrals and evaluated as functions of
.

It is helpful to define the new magnetic fluxsurface label
. (See Section 8.11.)
It follows from Equation (11.101) that at the Opoints of the magnetic island chain, and on the magnetic separatrix.
Hence, we can write


(11.155) 


(11.156) 


(11.157) 


(11.158) 


(11.159) 
where

(11.160) 
, and


(11.161) 


(11.162) 


(11.163) 
Here, the functions
,
,
, and
are defined in Section 8.11.
Note that the factor in Equation (11.157) is generated by the discontinuity in the function
(i.e., the discontinuity in the pressure gradient) across the
separatrix of the magnetic island chain [23]. If this contribution is omitted then the sign of the integral is reversed.
The values of the first three integrals are


(11.164) 


(11.165) 


(11.166) 
Figure 11.1 shows the values of the integrals and as functions of
.
Note that and are both only very weak functions of
.