346 THE BELL SYSTEM TECHNICAL JOURNAL, APR[L 1951 



SO that the integrals in (73) will not be altered if we introduce a constant 

 multiple of cot J(?? — <^) in each integrand. This enables us to replace the 

 improper integrals by convergent integrals, and find Poisson's inlegrals in a 

 form particularly well adapted to the network problem: 



1 r'" 



VeM = -TT [^e(»?) - ^.(<^)1 cot \{x^ - ^) (id, 



^eM = ^ I [VeW - VeM] COt \{d - <p) d§. 



It Jq 



(75) 



It may be helpful to engineers to note the similarity between these in- 

 tegrals and well-known integrals connecting gain and phase, which are of 

 course the real and imaginary parts of complex transmission functions. 

 Actually, the only essential difference is the shape of the contour on which 

 the relations hold — here a circle, as opposed to the imaginary /)-plane axis 

 for the gain-phase relations. 



Poisson's equations analogous to (75) may be found for points outside 

 the unit circle by separating the real and imaginary parts of the original 

 integral (67). The resulting integrals are convergent and there is no need 

 to modify the integrands nor to indent the contour. 



14. Use of the Inversion Theorem for Non-Circular Contours 



We have seen that in the it'-plane the interior function Fi{w) is not in 

 general analytic inside Ci , so that the inversion theorem cannot be used 

 directly. In other words, if Fi{w) has singularities inside Ci then Fi(l/w) 

 will have singularities outside Ci and therefore cannot be the exterior po- 

 tential Feiw). 



Thus, in general, we may have to use Poisson's integral to determine 

 the exterior stream function. The inversion theorem may still be applied, 

 however, if it is possible to separate the interior function into two parts, 

 one of which, Fa , is analytic inside Ci , while the other, Ft , is analytic 

 outside Ci . We write 



f[(w) = Fa(w) -f F,(w), (76) 



and note that Fa{l/w) is analytic outside Ci . Since Fb{w) is also analytic 

 outside Ci the exterior function is given immediately by 



fUw) = Fail/w) -f- F,iw), (77) 



where the transformation has to be applied only to Fa . 



This represents at times a real simplification of the charge distribution 

 problem, since Fbiw) is the same on both sides of the contour and therefore 



