A-2 



Take t to be a small sphere with radius e about the origin. Then the normal 



derivative is the radial derivative |^^ , and this is constant over the surface of 



o r 

 the sphere: 



V^cp- lim 3- (4tc ) 



4TTe 

 e-o 



9 cp 



3 ^^^ 



= lim — ^ ( - 1 + ik e ) 



r=e e e ^ 



e-o 



= lim / 3 -4n+4rrike 



4ne L J 



£-0 



(A- 6) 



-4tt 



Hence v cp blows up at the origin as , and the limiting volume integral of 



V^ cp over a small sphere about the origin will remain finite: 



lim 

 T->o 



I 



dT v= 



4 TT 



(A-7) 



Note that the volume integral of k tp over T does approach zero, even though 

 cp itself blows up. As a result, cp as given in (A- 3) does not satisfy (A-1) at the 

 origin, but satisfies the modified problem: 



V'^cp + k2 



4 n 6 (r) 



(A-8) 



where 6 (r) is zero everywhere, except at the origin where it becomes infinite in 

 such a way that [ dT 6 (r) = 1. Such a function has the property that J d T f (r)6 (r) = 



f (o) for any f. 



If the singularity is at a point r' rather than at the origin, a shift of origin 

 will show that: 



(r, r') 



e I r - r'l 

 4 ttI r - r'l 



(A-9) 



satisfies: 



V^ ilf + k 



(r - r' ) 



(A- 10) 



Arthur Sl.littleJnt. 



s-7001-0307 



