Leathem — Periodic Conformal Curve- Factor's and Corner- Factors. 49 



A corresponding specification of /(a) is that if the arc of any smooth closed 

 curve of perimeter 47r^ be 2'TrF{w), then F(2'rra/\) is a possible form of /(a) 

 11. While there is no general formula for expressing the integral 



J = - log sin j ^ (70 - a) j /'(a) dn, (38) 



a 



which occurs under the exponential operator in formula (32), as a function of 

 w, it will be shown that, for certain types of /'(a), the integral may be 

 evaluated by a method of contour integration. 



The first step is to indicate any selected value of w, as it appears explicitly 

 in the above formula, by ('_'„, and to replace the real variable a by the complex 

 variable w, whicli is to be the variable of integration ; it is to be understood 

 that when ?y is real it is to be the same as a. The change in the argument 

 of /' gives a function /'(vj) which is identical with /'(a) when w is real, but 

 which is otherwise a function of a complex variable, possibly possessed of 

 singularities which are quite foreign to /'(a). The integral which comes up 

 for consideration is 



- log sin ] — («'„ - w) [ /'(w) dw, (39) 



TT (A ) 



and the value of this, when taken round a suitable contour in the v: plane, has 

 to be examined. 



The contour found to be most suitable consists, in the main, of a rectangle 

 whose sides are in the lines 



<p = a, ij) = a + X, yfr = 0, yfr = t, 



where t is positive and may be made indefinitely great. There must, however, 

 be cuts from the boundary to infinitesimal circular cavities round all branch 

 points and infinities of the subject of integration, and it is convenient to take 

 for these cuts straight lines which start from the line -t^ = t and run parallel 

 to the line <j> = 0. The complete contour includes each side of each cut, and 

 the circumference of each infinitesimal circle. 



The point iv„ is taken inside the rectangle, and at this point the function 



-11 



TT, 



y'iioo- 1'-)} 



has a branch point. For tv - w,, small the singularity is sufficiently repre- 

 sented by TT"' log (Wo - iv), and it is seen that, if the point w describes an 

 infinitesimal circle round vj„ in the conventionally positive sense, a constant 

 2-i is added to the value of the function for each complete circuit. The same 

 is therefore also true for any other circuit round i'j„, whatever its size or shape, 

 provided it does not surround the other branch-points corresponding to Wq ± nX, 



