GUIDED-WAVK IMtOl'Ac; ATIOX TIIIIOICII (!^ !{( ).M A( ;\ imc MEDIA (101 



many features of the G^-coiitours are already determined by the position 

 of the contours (r = oc , aiid (/ = 0, across which G changes sign (from 

 ± X to T X or from ztO to TO). Because of their special role in the 

 sul)se(iuent analysis it is desirable to introduce a scheme for their enum- 

 eration. The infinity and zero curves in the right-hand half-plane will be 

 (lei Idled by / and 0, respectively, those in the left-hand half-plane by /' 

 and ()'. All but two of the /-curves arise from the poles j„ of F. Their 

 eciuations are 



1 — X .9,2 



1 - o-X 



= jn'/ro n = 1,2, 



Each of these curves has two branches, one in the half-plane X > 0, one 

 in X < and these are called /„ , /„' respectively. All /„ curves pass 

 through X = 1, cr = 1, all/,/ curves pass through X = —1, a = — l.The 

 lines X = 0, X = — 1 are also infinity curves to be denoted by I a , Is 

 respectively (As X -^ +1, (? tends to a finite value). 

 Zero curves of G are given by 



or in a more readily computable form by 



_ 1 _ /V(l - X^) 

 X \{F-\\)f 



(35) 



The branches of F ^(X) may be labelled according to the scheme: "0" 

 for - X < \F~\\)\- < J?; "1" forii' < [/^'(X)]' < J2 and so on. The 

 ?ith branch of F~\x) gives rise to an 0„ curve for X > and to an 0,/ 

 curve for negative X. All 0,/ curves pass through X= — l,o-= —1; all 

 save one of the 0,^ curves pass through X = 1, o- = 1. The exceptional one, 

 seen to be Oo , is associated with the "0" branch of F~\\) on which 

 F~ (1) = 0. For fixed <r, G tends to zero as X — ^ =o, hence the vertical 

 lines X = ± a= are also zero curves, to be denoted by 0^ and 0^' respec- 

 tively. 



In a sense the two branches of o-X = 1 are also zero curves, to be called 

 Of and Oc'. 0^ and 0^' are zero curves only when viewed from "one side." 

 In the right half-plane, for X < 1 as aX — > 1 — and for X > 1 as 

 crX — > 1 +0, the argument of /' tends to infinity and remains real. 

 Therefore G passes through all \alues an indefinite number of times and 

 (t\ = 1 is a limit hue of all contours, G = constant. For X < 1 as 

 o-X -^ 1 + and for X > 1 as o-X —> 1 — 0, the argument of F is 



