56 SAMPLED-DATA CONTROL SYSTEMS 
in F(p) will be enclosed. This condition results in an infinite number 
or a finite number of residues, depending on the choice. Using the 
right-hand enclosure, it is readily seen that the residues which result at 
the various poles described by (4.11) will be 
+o 
F*(s) = 7 D F (s - 7) (4.12) 
This expression cannot be reduced to closed form and is seen to be 
identical with (2.26), which was arrived at by a different procedure. 

| 
Poles of i 
/ 
i oe ee 
—e i (s—P) / 

Fic. 4.2. Contours used in evaluation of complex-convolution integral. 
If, now, the path of integration is closed to the left, it is seen from 
Fig. 4.2 that only the finite number of poles of F(p) will be enclosed and 
that there will be a finite number of residues to evaluate. If the various 
poles of F(p) are designated as p;, then the value of the integral is given 
by 277 times the residues at these poles: 
E*(s) = > res. Fo) a (4.13) 
poles of 
F(p) 
Now, if e”* is replaced by the auxiliary variable z, (4.13) can be rewritten 
