128 BELL SYSTEM TECHNICAL JOURNAL 



The form of the final expression has been chosen so as to be suited to the 

 use we shall make of it. 



Another double integral which appears in our work is 



/+C0 

 dvQaivo) 

 ■ CO 



(A2-10) 

 • / dv a(v) exp [ — ik(v + vo) — y \v — vo |] 



J_oo 



where a(v) is an even function of v and is such that all of the integrals 

 encountered converge. We begin our transformation by dividing the 

 interval of integration (— =o , oo) for z'o into (— co, 0) and (0, x). Making 

 the change of variable Vo = —vo,v= —v' in the first interval, dropping 

 the primes and using a{—v) = a{v) leads to 



j(k^y)^2f dvoa(vo) f dv aii^e^"^'-'"^ cos k{v -\- Vo) (A2-11) 



Jo J-oo 



We now split the interval of integration of v in (A2-11) into the intervals 

 (— oOj 0), (0, I'o), {vo , °o). In (— Gc, 0) we change the variable from v to 

 — v', drop the prime, and use a(—v) — a{v). By paying attention to the 

 sign of V — ro we may remove the absolute value sign. By changing the 

 order of integration in the double integral arising from the third interval 

 (in which < z'o < "^ , vq < v < oo ) we may show that it is equal to the 

 double integral arising from the second interval. Thus 



I{k^ y) = 2 f dvo a(vo) [ dv a(v)e""~""'° cos Hvo - z') 



+ 4 f dvo a(vo) f " dv a(v)e-"">^'''' cos Hvo + v) 

 Jo Jo 



When a{v), y and k are real we may write (A2-12) as 



I{k,y) = 2! r dv aiv)e-'"-''''\ 



I Jo I 



+ 4 Real C dvoa{vo)e-" "''''' f" dv a(v) e'"^'"' 

 Jo Jo 



and when 7 = ik we have 



I{k,ik) — 2 dvoa{vo) / dv a{v}e'^''"'' 

 Jo Jo 



+ 2 f dvoaivo) f " dv a(r)[/''^" + e-^"""]. 

 Jo Jo 



(A2-12) 



(A2-13) 



(A2-14) 



