since, for integral n, 



gi(.i-n)2n ^cos [(l-n)27T] = 1 



But if n = 1 the integral becomes 



277 



J z-a X 



[30a] 



Thus the residue of k{z -a) " is for n > 1 but equal to k for n = 1. Often /(s) can be written 

 in the form 



/(2) = 



{z-a)'" 



where m is a positive integer and the function g{z) is regular both at 2 = a and in its neighbor- 

 hood. Then g{z) can be expanded in a Taylor series near z = a: 



g{z) = Oq + flj (2 -a) + a„ (z -a)^ + . . . 



By substituting this series for g{z) and using 

 the results just obtained, it is seen that the 

 residue of f(z) at 2 = a is a , or the coef- 



' ^ ' m — 1 



ficient of the power {z-a)"'~^ in the series. 

 Or, the residue of f{z) o-t z = a also equals 

 ^(") {a)/n\ where g^"'^ {a) denotes the value 

 of the n derivative of ^ at 2 = a. 



Figure 33 - Integration around two 

 singular points Q, R. 



31. THE SCH^VARZ-CHRISTOFFEL TRANSFORMATION 



This transformation is useful in two-dimensional hydrodynamical problems that involve 

 boundaries in the form of flat surfaces, so that their trace on the a;y-plane is a polygon. It 

 may be an ordinary finite closed polygon, such as /I ^j A^, A. ^4, in Figure 34, or the bound- 

 ary on the ary-plane may consist of one or more broken lines each of which extends to infinity 

 in at least one direction. Boundaries of the latter sort can be formed out of a finite polygon 

 by allowing one or more vertices to recede to infinity and perhaps to spread out there; they 

 are often regarded as closed polygons with vertices at infinity. 



The Schwarz-Christoffel transformation maps the sides of such a polygon onto the real 

 axis in another complex plane, and maps the interior of the polygon into the upper half of this 

 plane. If the polygon has vertices at infinity, the space on either side of it may be defined 

 as the interior. 



57 



