DIFFRACTION OF RADIO WAVKS BY A I'AKMfOlJC ( ' V !.! NDKR 



70 



to the \-aliie of tlio integral is exp lj(/(i)] times 



2ir J 



exp 



- Z I'l^U - /o)'/'A-! 



(// 



~ (27r6..)~''-[l + ! -/>,/?, + lO^'B,} 



(10.4) 



+ 1-/^/^. + I35';i + oGb,h,]B, - 2\00hl(Mlh + 55(280)6356} 



+ •■•] 



w here Bk = {'Iboy'' / k! . The sign of (27r62)~^'" is chosen so that the argu- 

 ment of the right hand side of (10.4) is equal to arg (dt) at t = to on 

 the path of steepest descent. The derivatives oi f(t) at to give 



^2 = 2('o - /i)//c, b, = 4/1//0' , 64 - -12/1//,^ , 



-biB-2 + lO^afi;} = 



24/0(^0 - ^1)=^ 



The values of these quantities at the saddle point ^i may be obtained by 

 interchanging to and /i. If more terms of (10.4) are desired they may be 

 obtained from the formal result 



~ exp - E«a7VA-! 

 2ir J-x A-=2 



dt 



(27ra2) 



^ (10.6) 



1 + E y-^i^iO, 0, -aa, -«4, • • • , -a2.)/A-!(2a2)' 



where F„ (oi, a^, . . . , ««) is the Bell exponential polynomial." It is neces- 

 sary to rearrange the terms given by (10.6) in order to get them in 

 groups ha\'ing the same order of magnitude. A more -careful treatment 

 of the terms in the asymptotic expansions for Dn{z) has been given by 

 Watson." His method is similar to that used by Debye for Bessel func- 

 tions. 



In our woi'k we shall deal with two different complex planes, and the 

 reader is cautioned against confusing them. One is the complex /-plane, 

 shown in Fig. 10.1, which contains the paths of integration for integrals 

 such as (10.1). The other is the complex w-plane, shown in Fig. 10.2, 

 which is introduced because we are often more interested in Uu(<^), etc., 

 as functions of m = n -\- I than as functions of z. In the eai'lier sections 



" E. T. Boll, Kxj)()iipiiti:il Polynomials, .Vnii. of M.ith. 35, pp. 25S-27y, 1<W4. 

 The polvnomiiUs are tahulatod up to n = S hv John Hlordan, Inversion Formulas 

 in Normal Variable Mapping, Annals of Math. Stat. 20, i)p. 417-425, 1949. 



