POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 



345 



the integral may be written 



v.W + i^M^.,^^ 



(71) 



where P denotes the principal valuef of the integral, corresponding to the 

 contour C'\ that is, with an infinitesimal segment at the singularity ^ = (p 



\=eL<> 



Fig. 16 — Unit circle contour with semicircular indentation at p. 



omitted. In the integral (71) the value of VeW is known (since it is equal to 

 F,(t?) on C ) and we shall now show how to determine "^e from Ve. 



If we separate the real and imaginary parts of (71) we find 



VM = -^ P f [VeW + ^eW cot i(^ - ^)] d^, 



Jtt Jo 



'TT ^0 



,2)r 



"i^eM = ^ P [ [VeW cot i(^ - <p) -^e 

 ZTT Jq 



(72) 



W] dd. 



Since we have assumed that Feiw) vanishes at infinity the integrals of F« 

 and ^e round the circle will be zero, and (72) reduces to 



,2)r 



d^ 



VX<P) = -^P I ^e(t?) cot i(.? - ^) 

 ZTT Jo 



^.(<^) = ^P f VeW cot i(^ - <P) dd. 

 ZTT Jo 



Further it is easy to verify that 



/.2r 



P / cot iid - ^)dd = 0, 

 Jo 



t E. T. Whittaker and G. N. Watson, Modern Analysis, Cambridge, 1920, p. 75. 



(73) 



(74) 



