152 BELL SYSTEM TECHNICAL JOURNAL 



and we have 



J— 00 •/— 00 



(7-7) 

 O(a^) 



which may be neglected. The other integral suggested by (3-14) is 



-2J r— 3/2 ;•- , / -2icv -3 , 2 2r , 



e K K av — I e c 4a:K e dv 



■00 *'— 00 



= 2aK c~ /(I — ic) 



(7-8) 



When V > 0, 



(7-9) 



^ = e.=o + i f [Kie'"' -I- ae--'") - 1]' dv 

 Jo 



= i f {K-e'"" - \f dv + {){(x), 

 Jo 



— -— [x — tan~ X — c -\- tan" c] + 0(a), 

 za 



.-v = (k e "" — 1)% 2q: <it = .t(1 + x')" (/x 



In the integrals containing exp {—2v) as a factor, ^ may be taken to be lev 

 since the integrand becomes negligibly small by the time ./a' differs signifi- 

 cantly from (7-9). We have 



/— 2f r— 5/2 j-2 J I — 2f/ 2 iav ^ N — | 4 2/ , 4ar r, —2v\2 j 



e K K dv =^ e \k e — 1) k a {-ie — 2e ) dv 

 •'o 



= / e~^^ x~^ K^ a I6e^°'^ dv (7-10) 



Jo 



-00 



o / — i[x— tan li— c+tan ^c]la/ —4 i — 2\ j 



— 8a I e {x + X ) dx 



J c 



where the integrals containing e"-" and e"^" have been neglected since their 

 contribution is 0(a^). When a becomes exceedingly small the exponential 

 term oscillates rapidly and the last line of (7-10) is likewise 0(pr). This may 

 be verified by integrating by parts, starting with 



exp Y dx = ia x~~{\ + .r-)(/(exp Y), 



Y = —i{x — tan~'.T) /a 



The last integral which must be considered is 



