*:":? •'=:-r*«a> Ew . fcrWoK ^, 



P + (8) " "if C08 ~ 1 (,/k o } 

 P (s) - -r cos -1 (-a/k ) 



— If O 



(10.11) 



then 



P + (s) + P_{ 8 ) - Y 8 (10.12) 



for s in D, and P + (s) are analytic in R + with a certain behavior at 



Infinity which will be determined shortly. 



Firstly it is necessary to define the function cos -1 (s/k ) for 



complex s. If s/k is real, cos -1 (s/k ) is defined as the branch for 

 o o 



which cos" 1 (s/k ) - if/2 whin s/k - 0, i.e., cos" 1 (s/k ) lies between 

 o o o 



and ir when s/k is real and between ±1. Let 8 - cos" 1 (s/k ). Then 

 o o 



,V - e lg ± e- iB 

 s/k o - - 2 



and so , JL\ 



. (•'-"I'l 

 16 - 4*-*-r-8- 



- i.a 



m 



where y - (s 2 - k 2 ) T with the branch cute 3S already discussed. The 

 logarithm here is defined to have its principal value, i.e., In z is 

 such that 



-it < Im £n s< -Mr (10.13) 



with a branch cut along the negative real z-axis. Noy take s ■ 0; the 

 corresponding value of f is -ik and hence 18 ■ £n{±(-l)} ■ + ~r", sc 

 thac 8 ■ if/2 if we zhoose the lower bign. Keiice 



B--i ln\— ; — S-l (10.14) 



( L ?l 



76 



.-.-„: u,..^.,. .... ...-,.■ -: a ...„->.»— ...ji 



