REGENERATION THEORY 



129 



Notation and Restrictions 

 The following notation will be used in connection with integrals: 



^{z)dz = lim <>^{z)dz, 



(1) 



the path of integration being along the imaginary axis (see equation 9), 



i.e., the straight line joining — iM and + ■iM; 



f 0(z)dz = lim f p{z)dz, 



(2) 



the path of integration being along a semicircle ^ having the origin for 

 center and passing through the points — iM, M, iM; 



I ^{z)dz = lim I 0(z)dz, 



Jc -V-* ^ J-iM 



(3) 



the path of integration being first along the semicircle referred to and 

 then along a straight line from iM to — i^[. Referring to Fig. 1 it 



y 



-iMU— ^ Z-PLANE 



Fig. 1 — Paths of integration in the c-plane. 



will be seen that 



/-/■ = / 



(4) 



The total feed-back circuit is made up of an amplifier in tandem 

 with a network. The amplifier is characterized by the amplifying 

 ratio A which is independent of frequency. The network is character- 

 ized by the ratio /(/co) which is a function of frequency but does not 

 depend on the gain. The total effect of the amplifier and the network 

 is to multiply the wave by the ratio AJ(io}). An alternative w^ay of 

 characterizing the amplifier and network is to say that the amplifier is 



1 For physical interpretation of paths of integration for which x > reference 

 IS made to a paper by J. R. Carson, " Notes on the Heaviside Operational Calculus," 

 B. S. T. J., Jan. 1930. For purposes of the present discussion the semicircle is 

 preferable to the path there discussed. 



