Then Y (a) exiat6 as an analytic function of r at all points in nhe 

 cooplex s-plane for which the integral converges. The integral over 

 any finite range of x certainly converges (and y(x) is certainly inte- 

 grable near x»=0) so that the convergence is dictated by the behavior of 

 y(x) as x ■* + ">. There 



y(x) exp isx ~ exp[-(s + k )x] (2.4) 



wher»; s ■ s + i*»j» and the integral up to infinity therefore converges 

 if s + k . > 0. It may also converge for some other values of s, but 

 what can be guaranteed on the basis of the anticipated behavior of y(x) 

 as x ■* + » is that 



Y (s) is analytic in an upper half-plane 



(2.5) 



Moreover, Y (s) has non-growing algebraic behavior as |s| ■*■ °° every- 

 where within the domain R if y(x) J finite at x = 0+ or has an 

 integrabl* singularity at x » 0+, so that |v (s) | = 0(]s| ) say for 

 sotse A > as |s| -*■ °° along any radius in the domain R . If, as nay 

 occur in sons problems, y(x) has a non-integrabie singularity at x ■ 0+ 

 (that is, a singularity at least as strong as x -1 ) then y(x) has to 

 be regarded ss a generalized function whose generalized half-range trans- 

 form Y (s) is Jtill analytic in an upper half -plane R , but now can have 

 alK'braic growth as |s| -* m . 



By analytic we mean that Y^(s) is single-valued and has a uni s 

 derivative 



<***,: r-T-J- I 



