It may be seen from figure 3b that the first 

 derivative 6' does not vanish except if < o < 1/2^''^, 

 for which B' vanishes for 2 values of t. Equation (11a) 

 shows that the 2 points where d' =0, that is where the 

 phase 8 is stationary, are given by 



t^(<.) = [l±(l-8a2)'^V4<». (12) 



We have < t_ < t^ < <», with t_ = and t^ = <» for 

 a = 0, and t_ = l/2'''2 = t^ for o = 1/2^^^ The 2 

 points of stationary phase t _ and t ^ ate apparent on the 

 left side of figure 1 for a = .1, .2 and 1/2^'^. Figure 3c 

 shows that the second derivative 6" vanishes for one 

 value of t, say tg, if a > 0. Equation (lib) yields 



to(<») = (r-i/r)/2'^^ (!3) 



where T is defined. as 



r = {[l+(l+2<»2)'^2]/2'^2a}'''^ (13a) 



The value tg for which 6" = 0, and for which 9' reaches 

 it's maximum as figure 3b indicates, is a decreasing 

 function of a. We have t^ = «> for a = 0, tg = 1/2'^^ 

 for a = \/2^^^, and Iq = for o = ». Furthermore, 

 figure 4, where the functions tQ(o), t_(a) and t^(o) are 

 depicted, shows that we have < t_ < tg < t^ < °o for 

 0<, a < \/i}''^ 



Fig. 4 — The Functions t()(cr), t_^(o) and t_(o) 



In the vicinity of the points of stationary phase t ^ 

 the first derivative of the phase-function may be 

 approximated by the two-term Taylor series 



9'(t;<») ~ (t-t^)e;; -i- (1-1^)^6 ;:'/2, (!4) 



where the function t^(<») is defined by equation (12), and 

 e'^and 6™ represent the values of the functions 9"(t;o) 

 and 9"'(t;o) for t = t^(o). By using equation (12) in 

 equations (lib) and (lie) we may obtain 

 e; = ±2'^2a(!-8o2)'^2/[,+4„2^(,_8„2)l/2]l/2^ (,5a) 



Q'l =:p-2'''^^96<»''/[l+4o2±(l-8o^)'''2l'^2. (15b) 



in the neighborhood of the point tg, where 9" = 0, we 

 have 



9'(t;o) ~ e^ -^ (t-t^)^^"/2, (16) 



where the fimction tg(a) is defined by equation (13), and 

 ©Q and Gg" represent the values of the functions 9'(t;a) 

 and 9"'(t;o) defined by equations (Ua) and (lie) for t = 

 tQ(o). The funaions tg(o), t^(o) and t_(o), Q'^a) and 

 eg"(o), Q-^a) and e'Jo), Q"l(a) and 9"' (a) are depicted 

 in figure 5. The signs of the functions 9g, ©g", ©';^and 

 9!^' are readily apparent from this figure. The Taylor- 

 series approximation (14) and (16) are useful for devising 

 an efficient numerical method for evaluating the integrals 

 (1), (3a,b) and (5a,b), as will be examined in detail 

 elsewhere. 



\ 



\ 



\ 



1 



\""'~ 



^, 



\t + 







"-.y^ " "~ 





J>< 





\ s: ^ 







N ^ ^^ _ ^ 



-V *"" 







\ 



- 





1 



Fig. 5 — The Functions tg(o), t + (a), 9o(o), 0g"(a), 

 0;^(o) and Q±'(o) for < o < 0.5 



Asymptotic approximations valid for z = and x 



oo will now be obtained for the integrals (1), (3a, b) 



and (5a,b). These five integrals may be expressed in the 



form 



n<|>|((x,o) = lm(c^+ 1^^+ -I- c^- 1^1,- ), (17) 



where < k < 4, c^* are constants, and \i>^ are the 



