POTENTIAL ANALOGUE METHOD OF NETWORK SYNTHESIS 



319 



On the real frequency axis, therefore, we have 



« = i[F{p) + F{-p)] = even part of F, 

 I ip = ilFip) - Fi-p)] = odd part of F. 



Specifically we may write 



2a = 2a„ + Z loglp'J - />' | - E log | /»' - / 



\pm + P/ \pn + P/ 



(7) 



(8) 



where the singularities occur in paurs, one of each pair being the negative 

 of the other. 



(a) (b) 



Fig. 2 — A point charge in the potential plane; (a) at the origin, (b) at the point Zm . 



3. Logarithmic Potentials 



In two-dimensional potential theory we are really concerned with uni- 

 formly charged Une filaments whose potentials and intensities are the same 

 in any plane perpendicular to the axis of the filament. Hence, it is conven- 

 ient to speak of a point charge g in a two-dimensional plane (x, y) and re- 

 gard the plane as the plane of a complex variable, z = x -\- iy. The poten- 

 tial of a charge q at the origin in this plane. Fig. 2(a), is proportional to 

 the magnitude of the charge and to the logarithm of the distance from 

 the charge. 



F = — g log p + constant, 



(9) 



where the constant may have any convenient value. Note that we are using 

 arbitrary units of charge and potential; in a coherent system of electro- 

 magnetic units the logarithmic term would have a constant multiplier. 



