;74 BELL SYSTEM TECILXICAL JOURNAL 



K + iX = Z = s + 72co lop: h -^■4coM , di 



« Jo \^^^^T^ + Mr 



// 



= s 4- /■2co ]o^ — + 4a)(P + iQ), 



Z,o = 72a) loc; -7 + /-Ico/i I , cos .t't(/ 



(23) 



(24) 



= i2u>\og^ + 4a;(P + iQ). 

 P 



The principal steps in the derivation of equation (18) are given in 

 Appendix I. 



The new definition cf P + iQ is 



P + iQ = //i I -^=^ COS .v'rJr. 



Jo V?" -h / + /ur 



Replacing / by i"^ and assuming that v is a real quantity this is 



/^QO -p(/i'+:/'+ix')r 



P + iQ = fxz'-R ■ dr, 



Jo V'' ^ 1 + I^T 



where R is used to indicate that the real part is to be taken. 



The asymptotic expansion is easiest derived by expanding 

 1/(Vt^ + 1 + fir) into an ascending power series in t and integrating 

 termwise. 



l/(Vr-^ + 1 + Mr) = 1 - MT + (jJ.' - |)r^ - (m' - m)^' 



whence, writing /;' + y' + '-v' = n^ , 

 P + iC) = txl' 



COS COS 2B , cos 30 , 



M — 7^; — r (m- — 2) — —r ^ • 



cos 50 



(m^ - m)^3! + (m^ - ^^ + I) ^^4! 



, ., ...cos 69 . 



whence, separating the real and imaginary parts, 



P = 



^V r 



cos I /^ , .V COS 30 

 + (2m" — 1) - — ^ — 



+ 3(12m'-'-8m^-3)^^ + 



