BOUNDARY VALUE OF INFINITE SERIES 



189 



and make the point z converge along the boundary to the point z = I, 

 then since (5) can be wi-itten 



1 - e" e'« 



2 sin 



^ '(- + -) 



the value of the series becomes ih, where b is arbitrary. 

 Example 2. Consider Tannery's classical example: 



^ 1 + ^2 ^ (1 + 02)2 



(1 + z^r 



= 1+z^ 



(1 + zT-' 



First, let z be real, and let z converge to by the regular sequence 

 z" = a"/{n — 1), a being an assigned real number. Then the value of 

 the series at is 1 — e~"', or at this point its value is every number from 



to 1 inclusive. Thus the real function represented by the series has at 

 zero a shear (a vertical segment whose lower end is and upper end 1) 

 which is represented by the totality ((0, 1)). This point is an essential 

 singularity when z is complex, for if - converges to zero as 



z = J 



n—1 



the value of the series there is 1 — e-(«+'V3)j or the totality ((0,0°)). 



Numerous examples can be given in illustration, we content ourselves 

 with merely indicating the following familiar ones generally used in the 

 texts on analysis. 



1 



Example 3. 



E 



[(n+1) 2+1] {nz+D 



1 



nz-\-l 



