Ogilvie 



as its Fourier transform, and further define: 



orjx) = - 27ra„o(x). 

 Then the result in (2-61 a) Ccui be rewritten: 



A.] dk e''^''Ko(|k|r)a*^^k) ~ ^ ,r„(x) log r - i^ f „(x) . (2-6la') 



1 



-00 



where 



/-»00 



f^{x) = \ de (7;(e) log 2|x-| l sgn (x-e). (2-6la") 



*^-00 



By manipulating the full integral containing Kq, one can also show 

 that: 



J-C dke'*'''KJ|k|r)a* (k) = - ^ C" —SuliLiL-^ , (2-62) 



which is easily recognized as the potential function for a line distri- 

 bution of sources. 



Similarly, the other integrals can be interpreted in terms 

 of dipoles , quadripoles, etc. In particular, we see that for n = 1 

 the inner expansion of the integral reduces to the potential in two 

 dimensions for a dlpole. We may consider the variable x as a 

 parameter, and then we have a different 2-D dlpole strength at each 



X. 



The Sequence of Near- Field Problems . In the near field, 

 we can formcillze our procedure by making the changes of variables 

 already mentioned, r = eR, y = cY, z = eZ, then assuming that 

 differentiation with respect to R, Y, or Z does not affect orders 

 of magnitude. Instead of doing this, I shall slmiply retain the ordi- 

 nary variables, r, y, and z» and I ask the reader to recall that 

 differentiation with respect to any of these three variables causes 



a change In order of magnitude. Thus, for example, 9 ^/8r = 0(4)/€) 



In the near field. 



In cylindrical coordinates, the Laplace equations and the body 

 boundary condition can be written as follows: 



716 



