

Thus we ht.ve a function E(s), defined by (3.6) in R and (3.7) in 

 R_, these two definitions being identical throughout D, which is analytic 

 and of at most algebraic growth at infinity in the entire complex plane. 

 According to the extended version of Liouville's theorem, the most 

 general such function is a polynomial P(s), and hence the most general 

 solution to the W-H equation (2.26) is 

 G + (s) + P(s) 



Y^(s) - 



K.(s) 

 + (3.8) 



Y_(s) - K_(s) [C_(e) - P(s)] 



for any polynomial P(s). 



In most applications the polynomial is fixed in degree, and some- 

 times also completely, from considerations of "edge conditions" at x ■ 0. 

 The edge conditions determine the behavior of Y (s) at infinity in R . 

 This is seen most clearly if we go to infinity alcng the imaginary axis. 

 Let 8 ■ iu with u purely real and positive. Then 



f + (iu) - J y( 



(x) exp(-ux) dx (3.9) 



which is the one-sided Laplace transform of y(x). When u •*• + °° only 

 small values of x make any contribution to the Integral, and in fact 

 WATSON* a LEMMA [see, e.g., 9] states that under appropriate conditions, 

 the asymptotic behavior of Y (iu) as u + + " la obtained by insetting 

 in (3.9) the asymptotic expansion of y(x) as x ■* 0+ and integrating 

 term by term. The proceir cen also be used in an inverse fashion to 

 fiat the behavior of y(x) as x ■* 0+ by examining the behavior of Y (iu) 

 as u* +»[14]. Note, however, that the behavior of y(x) as x ■* 0+ 

 is NOT simply related to the behavior of the full- range transform Y(s) 



16 



J i J . -J'-riJ i i. . . i a» ,-r ,f ', l- B t-.ta >iifi.1ra-iMi 



