I 



F + (8) " ~ 2^is" j F(t) dt " 0(8_1) (1 ° 4) 

 aa |s| -*■• in R + , while ; :c;j t.-ajIi, 



F_(s) - + 2~ J F(t) dt - OCs" 1 ) (10.5) 



I as (8| +«ln R_. .;.... v-: . ::. -.:.. . 



Without going into fine details, (10.1) is proved by applying 

 Cauchy's theorem 



i 



to a contour lying within D and enclosing the point s in D (F(e) being 



I 



defined only for s in D in the first instance). The contour is then 



deformed to consist of a rectangle with \^j — *• as its lower side 



•< — '""> as its upper, and with the ends of these sides joined at 



infinity by short sides parallel to the imaginary axis. In the limit 



these short sides (of finite length, less than E 2 - e t ) make no 



contribution to the integral, so that 



Frs ) . _L. f Ii£l . _1_ f F(t) 

 F(S) 2iri J t - s dt 2Tri J T^l dt 



where now both paths of integration run from - » to + «. But then, 

 according to the basic theorem of complex variable analysis, the first 

 term defines an analytic function as s varies without crossing the 

 integration path, i.e., it defines an analytic function in ttui upper 

 half-plane above the integration path. Similarly, the second tuna 

 defines a function analytic everywhere below the integration path 



For the behavior at infinity we have 



72 



s^aaawsw^^ , r ,. ,^-^,_,-_,.^,...^_^ 



