REGENERATION THEORY 131 



The function will not be defined for .-v: < nor for |2|= co. As 

 defined it is analytic ^ for < .r < co and at least continuous for 

 X = 0. 



The following restrictions on the network may be deduced: 



lim ^[/(-/v) I exists. (BI) 



J{iy) is continuous. (BII) 



w{iy) = AJ(iy). (Bill) 



Equation (5) may now be written 



AG(0 = -^. fw(z)e^Ulz = x^. r w(z)e^^dz. (11) 



From a physical standpoint these restrictions are not of consequence. 

 Any network made up of positive resistances, conductances, in- 

 ductances, and capacitances meets them. Restriction (All) says that 

 the response must not precede the cause and is obviously fulfilled 

 physically. Restriction (All I) is fulfilled if the response dies out at 

 least exponentially, which is also assured. Restriction (AI) says that 

 the transmission must fall off with frequency. Physically there are 

 always enough distributed constants present to insure this. This 

 effect will be illustrated in example 8 below. Every physical network 

 falls off in transmission sooner or later and it is ample for our purposes 

 if it begins to fall off, say, at optical frequencies. We may say then 

 that the reasoning applies to all linear networks which occur in 

 nature. It also applies to other linear networks which are not physi- 

 cally producible but which may be specified mathematically. See 

 example 7 below. 



A temporary wave fo(t) is to be introduced into the system and an 

 investigation will be made of whether the resultant disturbance in the 

 system dies out. It has associated with it a function F{z) defined by 



/o(/) = ^. f F{z)e^'dz = x^. r Fiz)e^^dz. (12) 



7^(2) and /o(/) are to be made subject to the same restrictions as iv(z) 

 and G{t) respectively. 



Derivation of a Series for the Total Current 



Let the amplifier be linear and of infinite power-carrying capacity. 

 Let the output be connected to the input in such a way that the 



* W. F. Osgood, "Lehrbuch der Funktionentheorie," 5th ed., Kap. 7, § 1, Maupt- 

 satz. For definition of "analytic" see Kap. 6, § 5. 



