396 SUKFACE WAVES. [CHAP. IX 



Hence in (13) we must replace P by Q/ir.Sk, where 



and integrate with respect to k between the limits and oo ; thus 



If we put =& + m, where k, m are taken to be the rectangular coordinates 

 of a variable point in a plane, the properties of the expression (18) are 

 contained in those of the complex integral 



It is known (Art. 62) that the value of this integral, taken round the 

 boundary of any area which does not include the singular point (=c), is zero. 

 In the present case we have c = K + i^ , where K and /^ are both positive. 



Let us first suppose that x is positive, and let us apply the above theorem 

 to the region which is bounded externally by the line m0 and by an infinite 

 semicircle, described with the origin as centre on the side of this line for 

 which m is positive, and internally by a small circle surrounding the point 

 (K, /*!). The part of the integral due to the infinite semicircle obviously 

 vanishes, and it is easily seen, putting g-c=re l9 , that the part due to the small 

 circle is 



if the direction of integration be chosen in accordance with the rule of Art. 33. 

 We thus obtain 



r. 



7 *- 



* 



which is equivalent to 



* K+ ** 



dk ............ (ii). 



On the other hand, when x is negative we may take the integral (i) round 

 the contour made up of the line m = and an infinite semicircle lying on the 

 side for which m is negative. This gives the same result as before, with the 

 omission of the term due to the singular point, which is now external to the 

 contour. Thus, for x negative, 



/ e ikx r*> 



j : -dk= I 

 k-( K +l pl ) J 



dk (iii). 



An alternative form of the last term in (ii) may be obtained by integrating 

 round the contour made up of the negative portion of the axis of , and the 



