-■ Pien and Strom- Tej sen 



For later convenience, the range of integration is divided into two and Kj is 

 written 



Ki = Ki^ + Ki^ (33) 



with 



l_( I. i,£M),, ,., (34a) 



1 4-np^n Jg.^\ R3 R 



3R,M 



^^ ^-p^n Jq \ r3 



^ 1 dX . (34b) 



47rp, 



Similarly we also define K^ , K^^ and K^^, K^^. 



Evaulation of Kernel Functions 



The preceding integrations cannot be carried out functionally. To avoid a 

 long tedious numerical procedure, we approximate sin \ and cos k as follows: 

 Let 



\ = ka + y , 



where y varies from to a and k is an integer varying from to ^o. Then we 

 have 



cos X. = cos ka cos y - sin ka sin y , (35a) 



sin ^ = sin ka cos y + cos ka sin y . (35b) 



If a is chosen small enough, we can approximate cos y and sin y as follows: 



cos y = 1 + ajy + ajy^ , (36a) 



sin y = y + a^y 



2 , (36b) 



where a^, a^, and a^ are so chosen that the best approximation can be made 

 within the range of y. From Eqs. (35) and (36), 



sin \ = Sq + Sjy + Sjy^ , (37a) 



where Sq = sin ka, Sj = aj sin ka + cos ka, and S2 - 32 sin ka + a^ cos ka, and 



cos \ = c„ + c.y + Cjy^ , (^"^^^ 



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