380 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



morphic function, which term implies that the function is given by the 

 ratio of two entire functions, one or both of which is transcendental, and 

 that the singularities of the function are ordinary poles, except for the 

 point at infinity, which is an essentially singular point. In order to realize 

 the particular network developments to be given, it will be supposed 

 that the function satisfies the further restrictions given below: 



(1) All the poles lie in the left half of the p-plane ^^'ith none on the 

 imaginary axis. 



(2) F{p) = F(p). (The superbar denotes the complex conjugate of 

 the unbarred symbol.) 



(3) Real part [F(ii>))] > for all real values of o). 



These three conditions are necessary to insure that the function is the 

 impedance of a possible linear, passive electric circuit structure. Inter- 

 preted physically in terms of this possible equivalent structure, the first 

 condition specifies that the structure shall be stable ; that is, every natu- 

 ral mode of oscillation dies away exponentially. The second condition 

 specifies that the natural oscillations are real functions of time. The 

 third condition specifies that if a sinusoidal current flows at the driving- 

 point terminals of the equivalent structure, the average real power de- 

 livered to it will be positive. Since these three conditions, or their equiva- 

 lents, are frequently mentioned in discussions of network theory, it is 

 assumed that they are understood without more detailed explanation. 



In addition to the above restrictions on the form of the impedance 

 function, the following two conditions, while not necessary, will be im- 

 posed to limit the scope of the discussion: 



(4) All the poles of F(p) are simple. 



(5) F(p) = 0(1), exactly, as | p | — > m everywhere except at the poles. 

 Condition (4), while limiting the scope of the exposition required, does 



not restrict the application of the results in any important way, because 

 most impedance functions for which a network representation may be 

 required have only simple poles. 



Condition (5) implies that as p increases along any straight line drawn 

 through the origin and not passing through any pole of F(p), the modu- 

 lus of F{p) either approaches a limit or oscillates between finite Hmits. 

 The physical implication of this condition is that the response of the 

 network as a function of time to a suddenly applied cause begins ^vith a 

 discontinuity of the same degree as that of the cause. For example, the 

 current response of the network to an applied step of voltage begins with 

 a finite discontinuity. This behavior is a characteristic of continuous 

 (non-lumped) electromagnetic structures, which furnish the principal 

 application of the network developments to be described. 



