■?->i'V^V,»<,;:.--ifj*«-™»,»™ 



because exp(-isx) is then exponentially small at infinity in R and 



Y (s) is at least algebraically small there. (This result should really 



T 



be proven carefully using Jordan's lemma; see [7,13].) 



Correspondingly, if C is any contour from -°° to +» lying within 

 R and such that Y (s) exp(-isx) is intagrable over it, then 



y(x) H(-x) - — I Y (s) exp(-isx) ds (2.14) 



211 J c " 



Now if C i my contour from • ■•■*> to +°° lying everywhere within the 

 strip of D cf overlap between R and R , then we can identify C with C 

 for (2.13), C with C_ for (2.14) and obtain by addition of (2.13) and (2.14) 

 the inversion theorem 



y(x) - — I Y(s) exp(-isx) ds (2.15) 



2TT J C 



We remark here that our convection e.xp(-it;t) for the time factor 

 is consistent with the formulas (2.9) and (2.15), in tha sense that we 

 are rat.' t>" taking F.T. 's in space and time with the definitions 



t f 



Tf(s,ui) ■ J l yvX.t) exp(isx + iut) dx dt 



if JVC. 



y(x,t) | I Y(s,io) exp(-isx -ioit) ds du) 



(2u) r 



Note that if a time factor exp(+iti>t) were taken, and k ,k, were defined 



o > 



as k » uj/C , k, ■ to/C- , then we would have to give k ,k. small negative 

 O C * 1 o i — ° — — 



imaginary parts in order to secure a strip of overlap ir. which to take 

 F.T. 's in x. 



10 



*F*^^^w<^ ^a^<t:, ^ Hlfe a s M j 



