(A2-2) 



126 BELL SYSTEM TECHNICAL JOURNAL 



1 r 



Ii{u, v) = j \o^(ch u — cos 6) lo,:; (ch v + cos 6) dO 



IT Jo 



(A2-1) 



2 r 



Iz{u, i) = I COS 26 log {ch u — cos 6) log {ch v — cos 6) dd 

 ■w Jo 



Ii{u, v) = COS 26 log {ch u — cos 6) log {ch v + cos 9) dd 



TV Jo 



Assuming m and v to be positive and using the expansions 



00 



log(c^ u — COS 6) = log(e"/2) — 22^ w~'g~""cos n6 



n=l 



00 



log(cA u + cos <9) = log(e"/2) - 2^ (-)"«-'e-""cos w0 



n = l 



leads to 



oo 



/i(m, v) = log(e"/2) log(eV2) + 2X) ^-^e-""-"" 



n=l 



00 



/2(w, v) = log(e«/2) log(gV2) + 2X) (-)";;- V"-"" 



n=l 



/3(m, t)) = -e-2"log(eV2) - e-2''log(6V2) + 2^-"-" 



00 



+ 22 '^~K» + 2)-ie-"»-"''(e-'-" + g-^") 

 h{u, v) = -g-2"log(eV2) - g^-'log (e"/2) - 26-"^" 



00 



+ 2^ {-Yn-\n + 2)-ie-"»-"''(e-2" + g-^") 



n=-l 



When 2f or v are negative they are to be replaced by their absolute values 

 in the expressions (A2-2, 3). 



Now we consider the double integral 



(A2-3) 



/+00 - +00 



di'o I d-j 

 ■ 00 J — 00 



(A2-4) 



•exp [ — /i I To — r I —m \v — s\ —ic{v + 7'o) —8\v — Tq | ] 



in which n, m, c, 5 are real and positive and r and s are real. The double 

 integral may be reduced to a single integral by substituting 



—i\v—v 



e 



o' = ^ T" (6== + :t')-V"<"-''»^ dx, (A2-5) 



TT J— 00 



