As to conditions as x -+• ± » k we give k and q small positive 

 imaginary parts, 



k c - k, + ik 2 , q - q, + iq 2 (8.8) 



and then (8.7) gives 



4) - 0(exp - q 2 x) 



as x * +», y ■ 0. Away from y - we can expect that as x ■*+» <£ will take 



the form of an outgoing cylindrical wave, 



_ i 

 - r~ 2 exp(ik Q r) f(6) - 0(exp - k 2 x) (8.9) 



It then follows that all(+) functions linearly related to <}> and its 

 y-derivativea will be analytic in 



R + : im s > - min(q 2 ,k 2 ) (8.10) 



As x ■*—<*■, <f> contains an exponentially growing part which we have 

 split off in (8.4), so that <j> should behave like an outgoing cylindrical 

 wave, 



$ - 0(exp - k 2 |x|) (8.11) 



as x ■+• - «>. Then a.'LlQ functions will be analytic (and with algebraic 

 behavior at infinity) in 



R_ : Im s < + k 2 , (8.12) 



and the strip D is 



D: - min(q 2 ,k 2 ) < Im s < + k 2 . - (8.13) 



53 



