Van Wijngaarden 

 APPENDIX 



To invert the first term in (2.28) we consider the integral 



2^ J 



,St + AS/( 1 + S^) '• ^ 



dS 



S(l + S^) 



(Al) 



along a straight path a distance c to the right of the imaginary axis in the s 

 plane (Fig. Al). We deform this path to a closed contour as indicated in the fig- 

 ure. The contribution from the pole in 

 S = equals unity. To evaluate the con- 

 tribution from the branch points S = t i 

 we have to integrate around the branch- 

 cut indicated in the figure. We trans- 

 form now the s plane into a jj. plane by* 



(l + S^) 



(A2) 



Fig. Al - Closed contour 

 in the S plane 



Because of the branchcut we put the left 

 side of (A2) equal to +/i on the left side 

 of the branchcut and equal to -m on the 

 right side. The region between the 

 branch points in the s plane maps on 

 the imaginary axis in the /u plane: 



2\l/2 



Then (Al) becomes 



i 00 



1. 



1 



[Mt/(l-/i^)'''2] ^^^ 



d/j, + 



S 3 /x(1-m2) 



[Mt/(1-M2) 1 "^].-K^ 



d/x 



Introducing /x = ik this can be written as 



sin r\k + ^^ 



(l + k^) 



kt 



dk + 



(l + k2) 



dk 



*For this transformation the author is indebted to R. Timman. 



128 



