where T_ ~ -(K2 M : )/!1: I and T+ ~ -M .. /(K2 ) when I < M1.
i : the partial fraction expansion reads
R + 2 3 + 1, + ? .,(3
T -1T* 1 ( +T .)2 +1 (
with the RH i: 1 cendent, of T. Of course the RH are such that, I falls off at least as 1/T2
for large T, i.e., RI1 R2 R:,3/l R5/1 =- 0. identity is helpful for determining R5.
:: we have
dT '-. In(-T)- R2 n(-2 -) 3 n In (3 99)
oI l. 1 + 1 MMAl MI l
`'i.. I are given explicitly by
M,M (,i -)T (M, 1) 2 A A_
(T+ T-) (MIT+ + )2 )
I (MM 1)1' (M l- 1 A'
R2 j)j' r II ) -
( :~_-~- _1 ) (:A T_ M ) '
(3 100)
2( +I 11 4]) (3 101)
(3-102)
(3-11 :)
1[ (M.,i( _1)- lM,( C )- 1 ,
RT = --- ) -
-. -O M _
MiT++M Mi'
(3-104)
T ].!
T_-M
"hen the /'s are large, the sum over I can be replaced an integral over ( 1/'-' as
long as i is kept away from the endpoints 0, 1. We can isolate the terms that give rise
to singular end point contributions and simplify them considerably. We shall then -ate
the divergent contributions and ( :.1 them in detail.
First note that the worst enI. o:int divergence is c-1 n I near c 0 or 1/(1 C)
near 1. T::: we can drop all terms down by a factor of 1/ for small I or by
(MI 1)/ for I near Mi. I. for I < we note that ITM +A Mi ) ~ -K2 /. and