DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDER 493 



the path running from ^o to ^i, and Im f(ti) = Im /[^o exp ( — 2iri)] for 

 the one running from h to to exp ( — 27rt). The saddle points in order of 

 their height are k, h, U exp ( — 27rt), to being the highest. 



14. m = ip'/2. Fig. 11.6 (d). Here ^o = ^i and the dashed Hnes go 

 into the saddle point at ^i exp ( — 2Tt). The paths of steepest descent 

 change their directions upon passing through the saddle points. 



12. ASYMPTOTIC EXPRESSIONS FOR Un{z), F„(2), Wn{z) 



The asymptotic expressions given here are for z = I'^p and z = i~^'^p 

 [with i'' = exp (tV/4)] when n is not too close to z^/2. As mentioned 

 earlier, there is a close relation between our results and those given by 

 Schwid.* The main difference is that we regard n as variable and z 

 as fixed while Schwid regards z as variable with n fixed. Another point 

 of difference is that in place of the ?n = n + 1 which appears in our 

 expressions for to and ti the quantity n + 3^ appears in Schwid's work. 

 The quantity 2n + 1 appears to enter naturally when the asymptotic 

 values are obtained from the differential equations. This is seen when 

 the WKB method is applied to equation (9.7). 



By examining the paths of steepest descent shown in the figures of 

 Section 11 we can determine the saddle points corresponding to Uniz), 

 etc., (for z = i^ ^p) for various values of n. The contributions to the 

 integral (10.1) from the saddle points ^o to /i were discussed in Section 

 10. The contribution from the saddle point ^i exp ( — 27ri) (which enters 

 when z = t''p) is, from (11.6), exp {i2irm) times the contribution from 

 ti. 



Although we shall be concerned mainly with asymptotic expressions 

 for the parabolic cylinder functions themselves, expressions for their 

 derivatives may be readily obtained. Thus U'n{z) = dUn{z)/dz has the 

 asymptotic expression 



U'n{z) ~ 2/o [contribution of ^o to Un{z)] 



+ 2tx [contribution of ti to Un{z)] (12.1) 



+ 2/i [contribution of ^i exp ( — 27rz) to Un{z)] 



and similar expressions hold for V'n{z), W'n{z). These follow when Ave 

 note that differentiation of the integrals (9.1), which define the functions, 

 introduces a factor 2t into the integrand. Of course, if the path of in- 

 tegration does not pass throught a particular saddle point, its contri- 

 bution to (12.1) is zero. Upon replacing ^o and ti by their expressions 



* Reference 20, page 478. 



