K j (t-a^) (t-a^) . . . (t-a^) dt 



The integral is an improper one, but under the conditions assumed it converges at both limits 

 and has a finite value. 



For, in the first place, over a short range of t from some value t. up to a, , variation of 

 the remaining factors can be neglected and the corresponding part of the integral is nearly 

 proportional to 



/ {t-a^) dt={l-a^)-^ {t-a^) / [31b] 



Since by assumption a^ < 1, this integral is finite. 



In the second place, for large negative t the constants a , a„ . . . may all be dropped in 

 comparison with t. Thus the integral toward < = - oo reduces approximately to 



K / (t) dt = 2 't = 



[31c] 



which is finite since the sum of the or 's has been assumed to exceed 1. 



As t increases past a , the amplitude of < - a decreases from 77 to 0, as is clear from 

 Figure 35. Hence the amplitude of {t-a)~'*\ increases from -a, n^ to 0, and the amplitude of 

 dz/dt likewise increases by a n. Thus from t = a to ^ = (i„, 2 travels along another straight 

 line making an exterior angle 01 n with the first line. This line, too, is of finite length, as 

 illustrated by A^ A^ in Figure 34. 



It cannot be concluded immediately, however, that these two lines join at A.. For it 

 may not be possible actually to integrate past the point ^ « a,, at which dz/dt is infinite if 

 a is positive. To avoid this difficulty, we adopt the standard device of letting t pass above 

 a along a small semicircle centered at a., as illustrated in Figure 36. As t traverses this 

 semicircle, z cuts across from one straight line to the other, along a curve such as that drawn 

 near 4, in Figure 34. The change in z along this curve is given by the integral of dz/dt along 

 the semicircle. 



In terms of polar coordinates, as illustrated in Figure 36, on the semicircle 



t-a^ =T^ e , dt= ir^e dd 



since r^ is constant. For an approximate estimate, all other factors in dz/dt can be treated 

 as constants; let their product, multiplied by A", be denoted by Q. Then the change in 3 as < 

 goes around the semicircle is, from [31a], 



60 



