DIFFRACTION OF RADIO WAVES BY A PARABOLIC CYLINDER 489 



This is of the same form as (10.11). That /i exp ( — 27ri) is a saddle point 

 follows from (lifT(M-(Mitiation of the ofiuatiou 



lite^"") = /(/) + 2irimr - 2irmi. (ll.G) 



Combining (11.5) and (11. G) shows that the boundary between la and 

 lb is given by lUr = 0. This is indicated on Fig. 11.2. 

 We now examine the paths of steepest descent when m is small. Fig. 11.3 

 (a) gives a large view of all the paths, irrespective of arg m, when in/p 

 is small. 



4. m = O.OOop". Fig. 11.5 shows the vicinity around ^i. 



5. \m\ = 0.005p', arg m = t/2 -0.05. Fig. 11.3 (b). 



6. I w I = 0.005p", arg m = 7r/2. Fig. 11.3 (b). After passing through 

 ti the path encircles the origin clockwise and runs down into the saddle 

 point at t = h exp ( — 27ri). Since rrii is positive, (11.6) shows that 

 ti exp ( — 27ri) is lower than h. The path for arg m = 7r/2 — 0.05 sug- 

 gests that from h exp ( — 27rt) the path runs out to co exp { — iri) along 

 the path of steepest descent which lies directly under (on the Riemann 

 sheet for —Zir < arg ^ < — tt) the path which runs from ti to 

 t = =o exp (ix). It follows from (11.6) that, as t traces out a path of 

 steepest descent through ^i, t exp ( — 2iri) traces out a path of steepest 

 descent through ti exp ( — 2xf) directly under the path through ^i. 



7. I m 1 = 0.055p', arg m = 7r/2 + 0.05. Fig. 11.3 (b) shows that 

 after passing through ^i the path of steepest descent spirals in to f = 0. 

 According to (11.3), the spiral is given by 



r ^ (constant) exp ( — mrO/^ni) (11-7) 



when r is small and d large. Two things are to be noted. First, the type 

 of path is different from that for arg m = 7r/2 — 0.05. Hence arg m = 

 7r/2 marks a change of type similar to that shown in Fig. 11.1 (b), except 

 that here h exp ( — 27rf) takes the place of ^o. Condition (11.5) takes the 

 place of condition (10.11), and is satisfied by virtue of nir = when 

 arg m_ = ir/2. 



The second thing to be noted is that up until now all of the paths of 

 steepest descent have ended at ± «> and U, V, W could be deformed 

 into them without difficulty. How can Ave deform U, for example, into 

 a path of steepest descent when the path through ti spirals in to < = 0? 

 The way to deal with this problem is shown in Fig. 11.4 where U is 

 continuously deformed into two portions, one coinciding with the path 

 through ti, as shown in Fig. 11.3 (b), and the other with the path of 

 steepest descent through ti exp ( — 2iri). The second portion lies directly 

 "underneath" the first portion. 



