Falttnsen 
the new variable 
The next step is to change the integration eet so that the 
lower integration limit in the first integral is -¢ = a and the upper 
integration limit in the second integralis €e€ -(1-4-B) Then we expand 
the integrands. We can then write a two-term inner expansion of (61) 
as 
0 ai ie 
i 1KxX 
ane dk e 1 e*(c) (1, ) ) dk ee 
Got ie) > Bp 
Zo tBUr Ls 0 [te a Bey 
VW vw 
fe) Oo 
co 
v ye ne ikx 
a: dk e o * (k) 
2m Megs 
: i — ivx 
mag PE ig Sede o(v + k,) | dve” ¢(v + k,) 
e . 
het Teme ew eee ee ene Wey Pe 
2a 
0 
(D. 1) 
We should note that we have changed the integration limits 
from + ¢~'1-8-Blto +. This can be justified for the two first 
integrals, which is the lowest order terms, by using the fact that 
o *(k)k 3 remains bounded as k>+o. For the three last integrals, 
which is the highest order terms, it is obvious that we can change 
Vet a{{=ac 
the integration limits from f= e mt Big too 
1824 
