329-332] 



Multiple-valued Potentials 



275 



Conjugate Functions on a Riemann's Surface. 



331. An obvious case of a multiple-valued potential arises from the 

 conjugate function transformation 



W=<f>(z) (254), 



when <t> is not a single- valued function of z. Such cases have already occurred 

 in 317, 320, 323, etc. 



The meaning of the multiple- valued potential becomes clear as soon as 

 we construct a Riemann's surface on which <f> (z) can be represented as a 

 single-valued function of position. One point on this Riemann's surface 

 must now correspond to each value of W , and therefore to each point in the 

 TF-plane. Thus we see that the transformation (254) transforms the complete 

 TF-plane into a complete Riemann's surface. Corresponding to a given value 

 of z there may be many values of the potential, but these values will refer to 

 the different sheets of the Riemann's surface. If any region on this surface 

 is selected, which does not contain any branch points or lines, we can regard 

 this region as a real two-dimensional region, and the corresponding value of 

 the potential, as given by equation (254), will give the solution of an electro- 

 static problem. 



332. To illustrate this by a concrete case, consider the transformation 



W=z? (255), 



B 



a' 



-ft 



JT-plane. 



^-surface. 



FIG. 93. 



which has already been considered in 317. The Riemann's surface appro- 

 priate for the representation of the two-valued function * may be supposed 

 to be a surface of two infinite sheets connected along a branch line which 

 extends over the positive half of the real axis of z. 



To regard this surface as a deformation of the W -plane, we must suppose 

 that a slit is cut along the line OB (fig. 93) in the F-plane, and that the 



182 



