Falttnsen 
The poles of the integrand are important in the evaluation of 
I(k). They are given in the limit u—0 by: 
Ae (2v -k)k 
Let us first study the case in which these singularities are 
imaginary, which means k < 0 or k <2pv_ . Then we study case 
IL,@in whichy O:< ok =< Zev 
Gase lo kia 0 ork — Zane. 
By introducing a closed curve in the complex ( -plane pro- 
perly indented at the branchpoint i |v -k| of the integrand of (A-3) 
and using the residue theorem, we will get 
J Fae v/ 2 
‘ao vz - y Vk(k-2») 0 lvz pe) a 
re) = MBewioes tnd Seu Loada, fer eee eee mart 
2 Z 
Nice) i Ea 
-1vz v2 - be 
og ee 
Vl 2 
p2 -a -i (A -4) 
v-k : F 
Here a= It can be shown that the integral term in (A-4) 
is exponentially small with respect to e 
Caseul = O0— ke < 2)y)- 
The poles of the integrand of (A-3) are now real. The 
Rayleigh viscosity is helping us to determine how to indent the inte- 
gration path of I(k) around the poles. We get by using the residue 
theorem in the same way as for Case I that 
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