466 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



where F(m) = f(h) - f(to) + m log ^ and F'(7n) = log (to^/ti). Here 

 ^0 and ti are functions of m defined by (12.9). At Wi we have /oiS = 'i 

 and this leads to 



mi = -4iW(l + ^f, F(mi) = 2t/i(l - ^)/( 1 + 0), 



F"{nH) = -(1 + /3)/mi(l - /3). (7.40) 



When we attempt to deform the path of integration ACE of the 

 second integral in (7.38) into a path of steepest descent, we encounter 

 no trouble near Wi in regions I' a and I'h. The path passes through Wi 

 with arg {dm) = — r/4. Soon after passing through mi the path of 

 steepest descent strikes the boundary between I' a and //' at a point 

 we shall call G. At G the imaginary part of m is 2h{l - ^)/(l + /3) log /3. 

 The asymptotic approximation to the integrand changes its form at 

 this point. The choice of the path from G out to °c is not important 

 since it contributes little to the value of the integral. However, if we 

 insist on following paths of steepest descent, it turns out that we must 

 split the path of integration at G. 



When h}'^{l — j8) ^ 1, it may be shown that the value of second 

 integral in (7.38) is nearly equal to 



-2Mr[-2ir/^F"{mit" exp [F{im)] 



and this gives the second term in (7.36). 



So far, in this section, we have been dealing with the case of horizontal 

 polarization. Since the work for the case of vertical polarization (in 

 which H plays the role of the wave function) foUow^s much the same lines, 

 we shall merely list the formulas corresponding to those already ob- 

 tained for horizontal polarization. Mi, 13 and Mo, g are still given by 

 (7.4) and (7.20), respectively. 



S,ih) = Ml ~. Tr^rT^ ^^ + 0{h'/r"-), (7.41) 



Jl-. sm wn WnKZo) 



SM = -5,(0), (7.42) 



00 



Ssih) ~ 2iMr Z i3" 'Vn{zo)/'Wn{zo), fi < 1 (7.43) 



SM) ^2imA((s - ly' - ttJ 



t'"/3" 'VniZo) 



_sin irn d'W,Xzo)/dn_ 

 /3 > 1 (shadow region), 

 ris = sth zero of 'W„{zo), 



(7.44) 



