342 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



We now introduce the theory of conformal transformations, and use the 

 fundamental property that any simple closed curve in the finite />-plane 

 may be mapped, by an analytic transformation, on the unit circle in a second 

 complex plane, which we shall denote by the w-p\a,ne. Suppose that 



p = T{w) (62) 



is such a transformation. Primarily the transformation must be such that 

 points on the contour C in the />-plane become points on the unit circle, 

 Ci , in the 2£^-plane, but this is not sufficient to define T uniquely. To make 

 the definition unique, in a way which we shall find convenient in solving 

 our potential problem, we impose the following conditions: 



1) T(w) maps Ci on C 



2) T{w) maps the exterior of Ci on the exterior of C in a one-to-one 

 analytic manner ,^^s 



3) The point at infinity in the 7£;-plane corresponds to the point at 

 infinity in the />-plane 



4) r(+l) is real and positive. 



Now if our assigned transmission function in the />-plane is 



Fiip) = Viip) + i^iip) (64) 



the assigned transmission function in the ze;-plane is 



F'iiw) = Fi\T{w)] = V'iiw) + i^'iiw) (65) 



and our problem is to find the exterior function Fe{w) in the w-plane. Un- 

 fortunately this problem cannot usually be solved by the simple inversion 

 theorem for the circle in the />-plane, because the transformation (62) intro- 

 duces singularities in F'i{w) which are in addition to the singularities due to 

 the poles and zeros of the original function. The second condition of the 

 set (63) requires that Feiw) must be analytic outside Ci , but in general 

 Fi(w) is not analytic inside Ci and the inversion theorem will therefore not 

 lead to an analytic form for F'e{w). The second condition of the set (63) was 

 deliberately chosen to make the mapping Feiw) of the unknown exterior 

 function Fe(p) analytic outside Ci . The extra complexity of the potential 

 problem for the general contour C, as compared with the circle in the />-plane, 

 arises because it is not usually possible to define the transformation in such 

 a way that, simultaneously, the mapping Fi{w) of the known interior func- 

 tion Fiip) is analytic inside Ci . Two exceptions are when Fi{p) is constant 

 so that F'i{w) is also constant (the equipotential distribution), and when 

 T(w) is a linear function (when the original contour in the p-plane is also 

 circular). 



