2. As an alternative, m factors may be employed in the expression for dz/dt, with a 's 

 representing all of the external angles. In this case only one integral to infinity is obtained, 

 representing the length of one of the m sides, at some point of which ^ = c« occurs. The a's 

 are again subject to ot - 3 conditions; but here their number is m. Hence in this case three 

 of the a's can be chosen arbitrarily. 



Infinite Polygons. A corner of the polygon can be displaced to infinity in either of 

 two ways. 



1. If - 1 <^ a + a . . . <*„<!» the integrals to i = + «> and from t = - <x. no longer con- 

 verge, as is illustrated by Equation [31c], Thus the broken line extends to infinity at both ends. 

 The integral along the semicircle at infinity, in Equation [31d], also no longer vanishes. 



If «, + a2 + •••«„ = !» a„+ 1 = 2 - (aj + . . . a^ = 1 also, and the first and last 

 segments of the broken line, on which t < a, or t > a , respectively, differ in direction by 

 a , , TT = 77 and so are geometrically parallel. In this case [31a] can also be written 



9^=-K (a,-t) ... (a -0 



dt ^ I > y n I 



Thus on the first and last segments dz/dt has opposite signs, so that these segments are 

 traced in opposite directions. For their distance apart, measured from the last to the first, a 

 fresh evaluation of the integral in Equation [31d] gives 



n 

 As ^i K f 



in K 



Here the factor i causes \z to be perpendicular to both segments, whose directions are those 

 of + K. This case is illustrated in Figure 38a. 



Examples in which - 1 < a + a . . . «„ < 1 are illustrated in Figures 38b, c, d. Here 

 As, estimated as in Equation [31d], is infinite. In Figure 38d the geometrical polygon is re- 

 duced to a single semi-infinite line and its "interior" includes all the remainder of the s plane. 



2. As an alternative, one of the oi's may itself exceed 1. Then the integrals up to the 

 corresponding point a, as in [31b], diverge, and both adjacent sides extend to infinity. No a 

 should be made greater than 2, however. Two cases are shown in Figures 38e and 38f; in 

 38f the polygon consists of two unconnected infinite lines. 



As with finite polygons, a given infinite polygon can be transformed into the real t 

 axis in different ways. 



In any case, as t traverses its real axis positively, the upper half of the <-plane lies 

 to the left; hence, as explained under conformal mapping, the corresponding region on the 

 2-plane lies to the left as z traces the perimeter of the polygon. In the case of finite polygons, 

 the region on the left is the interior; with an infinite polygon, the region on the left is that 



64 



