(s + k )(s + ik,) (s - k.)(s - ik.) 



V> - T a.k'xs^kS ' K » - (s-k)(s-ikS < 6 ' 15 > 



Then division by K_(s) (K_(s)j«0 in R_ and soj*0 in D) gives 



,\ , % 1f_(s) Qi(s) + Q (s) 



V 8) V 8 > + KlsT ■ (s+k)(s+ik)(s-k)(s-ik) (6 - 16) 



and we have to make an additive split of the right hand side. Now 

 because of (6.12), Q (s) must contain the factors (s + k ) (s + ik ), 

 while Q (s) must contain the factors (s - k ) (s - ik ) so that we 

 can write 



Q (s) = (s + k )(s + ik )(a s + b ) 



n O O O O 



Q (s) = (s - k )(s - Ik )(a s + b ) 



where the coefficients a , a, , b , b, are known when any particular 

 o l o i 



set of co:ditions is specified at x ■ C. Now the right side of (6.16) 

 has the form 



a s 4 b 



(6.17) 



(s - k,)(s - ik t ) (s + k Q )(s + ik Q ) 



which is already in the desired form G_(s) + G (s). The reason for 

 this is that tnalyticity arguments have already been used to remove 

 pole singularities where they are not permitted and because pole 

 singularities are the only kinds of singularity which are present in 

 these one-dimensional problems this effectively means that the additive 

 split must already have been carried out. 



29 



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