Evaluation of Integrals
Figure: 8.1
The integrands
,
, and
.

In order to evaluate the integrals (8.109)–(8.111), it is helpful to define the new magnetic fluxsurface label
.
It follows from Equation (8.55) that at the Opoints of the magnetic island chain, and on the magnetic separatrix.
It is easily demonstrated that


(8.112) 


(8.113) 


(8.114) 


(8.115) 


(8.116) 
where
are standard elliptic integrals [1]. It follows that

(8.119) 
for , , , where


(8.120) 


(8.121) 


(8.122) 


(8.123) 
and use has been made of the easily proved result
.
The integrands
,
, and
are shown in Figure 8.1.
Note that
has a logarithmic (and, therefore, integrable) singularity at the island separatrix ().
The values of the integrals themselves are