942 Calculation of Be ssel Functions of Imaginary Argument. 



In terms of the old notation, therefore, on reduction, the 

 limiting values become 



x \ y/x* -t m 8 -$m log — - = - [ 



T = .r{(.r + m 2 )-l-\ 2 (.f« + m 2 )-S+...} . . . (12) 

 where the coefficients are defined by 



A, =-i, X 4 = *(27-9Gm 2 ), 



A 6 = JL (4640m 2 - L125-640m 4 ), 



J i s + 3 )\„+ 3 + {s + 2) 8 X frH + 2m 2 ,- . + 1) (6- + 2)X»_ 1 



+ m 4 *.(* 2 -4)\,_3 = 0, . (13) 

 and the identity 



1 +//,•'• + /<,.'- + • • • = (i+A2tf+M^+ • • 0" 1 - • ( 14 ) 

 The Limiting Forma of the substitutions (8) become 



Lt n-»P?( cosh -) = Lt f.^-)V, 



Li / .-'v"-Q:(cosh-)= Lt (£-)**-*, 



where Q™ has now been rejected in the first substitution, as 

 proportional by the second to e~ t which is very small, for only 

 moderate values of x* + m 9 9 in comparison with e*. 



Finally, therefore, by the use of (5) and (7), we obtain 

 the results 



I 



»-(c) 



(15) 



where 



T = ^{(^+m*)-i-X 2 (^ + m 8 )-i + A 4 (^ + m 2 )-i- ,,,J, 



Z = "1 1— M2 — i A^4 ( — -t— ) . .- f X 



1 ??ia??i \mam/ j 



x|(,,M-^)^-imlog ( ^ + ro ^_ m j, (16) 



and the coefficients are given by (13, 14). 



