we have 



(s 2 - k 2 ) Y (s) - 4 y'(O-) - isy(O-) (7.2) 



o - 



aid in order to avoid having a pole ct e •« -k eR_we must hav» 



+ y*(0-) + ik y(O-) - (7.3) 



Also as a check, as |s| ■* » in R_, 



Y_(s) - - - y(O-) 

 which ve also get from 



Y_(s) - J y(O-) + ...) e iSX dx (7. A) 



using Watson's lemma. 



The same differential equation holds in I < x < ». Define 



Y J _(s) « I y(x) exp isx dx (7.5) 



+ h 



y(x) being the Lotal displacement in 9. < x < °°. Then Y (s) is analytic 

 in the upper half-plane 



R, : Im s > - k . 

 + oi 



because of the exponential decrease, exp(-k .x) of y(x) as x ■+• + °°. 



ol 



However, Y (s) does not now have algebraic behavior at infinity in S. . 

 To find the behavior we write 



35 



