networks for impedance functions 381 



mittag-leffler's theorem^ 



Lot the polos of tlio given function F(p) bo Pi , P2, Vs ' ' ' , where 



< I Pi I < I P2 I < I p. I • • • 



and let the residues at the poles be Ai ,A2,Az • ■ • , respectively. Suppose 

 that it is possible to draw a sequence of closed contours, r„ , such that 

 C„ encloses pi , p2 , • • ■ Pn but no other poles and such that the minimum 

 distance of C„ from the origin tends to infinity witli /;. Suppose also that 

 F{p) satisfies conditions (2), (4) and (5) above. Then Mittag-Leffler's 

 theorem gives the follomng series development for F{p) : 



F{p) = F(0) + Limit i: ('_A_ + ^ (1) 



AT-oo „=-Ar \P — Pn Pn J 



The notation here used employs the con^'ention that 



p_„ = Pn and A_„ = A„, 



since, by virtue of condition (2), the poles occur in conjugate complex 

 pairs. The value, n = 0, then allows for a pole on the negative real 

 axis. 



Given any suitable function, the procedure is to determine its value 

 for p = and the location of its poles. The residues are next determined 



by 



An = Limit (p - pn)F{p). 



V-*Pn 



Then the Mittag-Leffler expansion can be written down at once. 



network representation 



In the series (1) the terms occur in pairs with conjugate complex poles 

 and residues. The object is to obtain a network representation of each 

 such pair of terms. If F(p) is taken as an admittance, the branches rep- 

 resenting the pairs of terms will all be connected in parallel; if F(p) is 

 taken as an impedance, they \vill all be connected in series. 



Methods for obtaining a network representation for a rational func- 

 tion, such as the one comprising a pair of terms in the series (1), are well 

 known. It is only necessarj^ to describe certain procedures of particular 

 application to the present problem. Brune has stated that the necessary 

 and sufficient condition for a network representation of a rational func- 

 tion of p to be realizable is that it be a "positive real function," that is, 

 a function that is real for real values of p and whose real part is positive, 



