334 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



sides of C. Then the potential will be continuous across C while the stream 

 function will be discontinuous by an amount which is determined by the 

 charge on C in accordance with equation (24). If we write 



Fiip) = Viip) + i^i^iip) (34) 



for our known function inside C, and a corresponding expression 



Feip) = V,{P) + i^eiP) (35) 



for the complex potential outside C, then Ve is determined by F* , while 

 ^e will be known if we know both^i and q. Conversely, q will be determined 

 if we know both ^i and ^e . Thus the problem of determining the charge 

 distribution on C may also be formulated as the problem of determining 

 the exterior stream function "^'e . To make the function ^e unique we specify 

 that it must be analytic outside C, and must vanish at infinity at least as 

 l/p, except perhaps for a logarithmic term which corresponds to an equi- 

 potential charge density on C. If the net charge on C is zero^e must vanish 

 at infinity. 



Thus, if it is possible to solve this potential problem we have a corre- 

 sponding solution of the charge distribution problem. The existence of a 

 solution has been proved, and is known as Dirichlet's principle, but its 

 solution has been formulated analytically only for circular contours. How- 

 ever, for circular contours in the />-plane simple methods of determining 

 ^e are available, and we shall discuss these before giving the general solution. 



10. The Power Series Solution for a Circular Contour 



When the interior transmission function is given as an analytic function 

 inside a circular contour, the exterior function may be determined by vari- 

 ous methods. An elementary method is based on power series expansions. 

 Since any analytic function of p can be expanded in a power series inside 

 a certain domain of convergence the method has quite general application. 

 To obtain the best form of power series applicable to our problem, we shall 

 start by considering the expansion of the complex potential for a given set 

 of lumped charges g„ located on the circle at points pn , 



F{p) = constant - S ^n log (p - pn). (36) 



n 



Inside the circle we have \ p\ < pniox each of the charge points />„ , and 

 therefore each of the logarithmic terms may be expanded as convergent 

 series in p/pn • Hence 



Fi{p) = constant - Z) ^n log (-/>n) - I] ?« log ( 1 - 7- ) 



n n \ pn/ 



= constant - YL qn\ - — - '^Tji - • • • \^ 



