DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDER 501 



where we have used 



t, = z/2 = i"'p/2 = {m,/2f" (13.7) 



and have arranged the terms within the brackets so that they represent 

 the first two terms in the expansion of 



m m . m , 



(27r)"^ / p/ W + 1 



(13.8) 



about niQ . 



The paths of steepest descent in the ^plane when m = rrio are shown 

 in Fig. 11.0(d). The three branches start out from I = t-i in the directions 

 arg (t — /s) = 15°, 135°, and — 105°. In this section we take the paths of 

 integration to be those of Fig. 11.6(d) even when m is not exactly- 

 equal to 7??o . Since we are dealing with asymptotic expressions we may 

 confine our attention to the region around t = (2 where the paths of 

 integration are essentially straight lines [the contributions from to exp 

 (— 2x1) are neghgible]. 



When (13.6) is set in the integral 



(l/27r0 / exp [/(/)] dt (13.9) 



we see that the initial directions of the branches are such as to make 

 {( — Uf/z positive (arg z = 45°). Some study of (13.6) and of the Airy 

 integrals we intend to use suggests that we change the variable of inte- 

 gration from t to s and introduce the parameter h where 



^-to = s(z/4:y'\ b = (m - mo)(2/zy" = (m - m,)/m,"\ (13.10) 



This and (13.6) converts the integral (9.1) for Vn{i'^p) into 



y.(,"V) = ^^/^V^,. f 



i(2.)-r(^i )•'-«><■="" (13.11) 



exp [ — bs — s' /3 + • • ■] ds. 

 When we use the Airy integral defined by 



.4/(0;) = T M cos {xt + t^/Z) dt 

 Jo 



= {i"/2Tr) f exp [-r^"xs - sV3] ds, 



Jooexp (i27r/3) 



(13.12) 



we obtain 



/ /A\113 22/2 r, 



T/ r -1/2 N U/4j e Zt -f.Ws 



V„{l p) '^ ; --TT- Trr, Al{bl ) 



i(2ry-r['^Y 



« ' '' (13.13) 



