320 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



For present purposes this would merely lead to a complication of the 

 argument. 



If we introduce polar coordinates, z = pe^^, we may consider a complex 

 potential 



W = — 5^ log z + constant = — ^ log p — iq<p + constant. (10) 



The real part of this function is the potential and the imaginary part is 

 the stream function. If the charge is at a point Zm , other than the origin, 

 Fig. 2(b), the corresponding complex potential is 



W — —q log (z — Zm) + constant. (11) 



For a set of point charges the total potential is simply the sum of the 

 individual potentials, 



W = —z] 9m log (z — Zm) + constant, (12) 



while for a continuous distribution of charges over a contour (C) the sum 

 is replaced by an integral, 



w = - I e(r) log (z - r) Mrl , (13) 



where \d^\ is an element of length on the contour. 

 In general we write 



W =V + i^ (14) 



where V is the potential and^ the stream function. We note that W in (12) 

 is analytic everywhere in the finite part of the z-plane except at points 

 occupied by the charges. Similarly, W in (13) is analytic everywhere except 

 on the contour (C) and at infinity. 



We may use the theory of analytic functions of a complex variable to 

 obtain various properties of the potential and of the stream function. First, 

 we remark that the derivative of W is unique, and may be written in either 

 of the forms 



dW dV , .d^ d^ .dV ,,^, 



dz dx dx dy dy 



whence V and ^ satisfy the Cauchy-Riemann relations, 



d^ dV d^ dV ,,^. 



dx dy dy dx 



The stream function and the potential are not independent; either is deter- 

 mined by the other except for a constant. 



The components of the electric intensity are obtained from V by the 

 relation E — — grad Y . Thus we find various alternative forms, 



