DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDER 473 



Incidentally, if, instead of taking the point C of Fig. 6.2 to be at 

 — 1 — ih, we take it to be at —J4 ~ '^h (a choice which receives some 

 support from the Airy integral representation obtained from the view- 

 point of the differential equations discussed in the first part of Section 

 13), the approximate integrals of (7.17) and (7.46) may be integrated 

 directly when h = and (3 = 1. It is found that 



Sn ^ -\-iM i/2, 



^33 ~ -iMi/2, 



and these add to gi^•e the values *S2(0) '^ iMi, SsiO) '^ —iMi required 

 by the half-plane case. 



It is seen that a rather thorough investigation of the errors introduced 

 by our approximations would be required to resoh'o the discrepancy 

 between (8.17) and (8.18). Since we do not intend to go into this subject, 

 and since the errors we have made may be as large as the }4 which ap- 

 pears within the square brackets of (8.15) and (8.16), we shall "split 

 the difference" between the two polarizations and omit the ^2 alto- 

 gether. This is done in Section 2 where t ~ Qp . 



9. THE FUNCTIONS U n{z) , V n{z) , W n{z) 



The functions Un{z), etc., are defined for all values of z and n by the 

 integrals 



Un(z) = J-. f r"-e-'^+^^' dt, 

 Zirl Ju 



VM = ^. t n-\r^'^'-Ut, (9.1) 



Zirl Jw 



where the paths of integration U, Y , W in the complex ^plane are shown 

 in Fig. 9.1. The cut in the ^planc runs from — ^ to and has been 

 introduced in order to make the function /~"~ one-valued. In some of 

 the later work the paths of integration will cross this cut. Of course, 

 this requires close attention to arg /. 



The initial and final points of the various paths (denoted in Fig. 9.1 

 by the subscripts i and /) are located at infinity. Arg I = — tt at Ui 

 and Wf and + tt at Vf and Vi. 



We shall give a summary of the properties of the functions (9.1) 



