corresponding oij, . . . a „■ When t is real and t < a, t - a is a negative real number. As t 

 explores the upper half plane and comes down to the real axis where t > a, as illustrated in 

 Figure 35, the amplitude 6 of t — a decreases by n. Let 6 be so chosen that 



2 =2 



Then, for the values of t under consideration, 



varies continuously between and r, and 



(t-a)~ ** " is a continuous function and is 



differentiable by the ordinary rule. Actually, "^ 



t may be allowed to go anywhere except to 



Figure 35 - Path past a singular point. 

 t = a, but it must not cross the vertical Line 



extending downward from t = a. 



Let all of the points t = a^, a^ . . . a^hQ treated in this manner. Then dz/dt and the 

 function z{t) obtained by integrating dz,/dt will be regular functions of t above and on the real 

 axis except at the points t = a., a . . . a^. 



Finite Polygons. Let a , a, . . . a„ be such that 



-\^ot.<\, y = i, 



The meaning of the first statement is that all of the a 's lie within the limits specified. 



Under these conditions, the transformation defined by Equation [31a] transforms the 

 real axis of t into a closed polygon on the 3-plane, To show this, the equation must be inte- 

 grated along the real axis. 



As t advances a distance 8t along its real axis, the s-point on the 2-plane undergoes 



a displacement 



dz 

 5z =— 8t 

 dt 



Since 8t as a vector is directed toward < = < + «>, the direction of 8z will make with the real axis 

 on the 2-plane an angle equal to the amplitude of dz/dt. This amplitude is in turn the sum of 

 the amplitudes of the various factors in the right-hand member of Equation [31a]. 



_ 01 



So long as < is to the left of a , the amplitudes of all factors such as {t-a) remain 

 constant, and so does amp {dz/dt). Thus, as t moves from - <» up to a^, z moves along a 

 straight line, as illustrated, for example, by A^^ A^'\n Figure 34. 



The total complex length of this line is 



59 



