■ a 



^*VWir< 'SKOWVU^M-; v'.s. 



The additive split of (s 2 ) 2 can be found by taking the limit of 



(10.11) as k •*■ 0. Define fcn.s to be the branch of Jin s with a cut from 

 o + 



0-0i in the lower half-plane, and with £n s real and positive when a is 

 real and greater than 1. Define £n_s similarly except that it has the 

 cut from + 01 in the upper half -plane. Then we can see that 



in.a - fcn_s - if Res > 

 J,n + s - £n_s - 2iir if Res < , 

 and both of these are covered by 



to + s - £n_s - 2iTT H(-Res) 



where H(x) ie the Heaviside function equal to 1 or according as 

 x > or x < 0. Since sgn x - 2H(x) - 1, we can write this now as 



Hn.s - in s » in - in sgn Res 

 and multiplying by s and using (10.21) gives 



2\? 



8 in .8 - s £n s ■ lira - in (s ) 

 Now define 



and then it follows from (10.23) that 



(10.23) 



(10.24) 



(s 2 ) 2 - P + (s) + P_(s) (10.25) 



and P + (s) is enalytic in R + (Im a > U) , P_(s) is analytic in R_(Im s < 0) , 



80 



