coaplax 



k + k + ik J v 

 o or oi | 



{ (2.2) 



k . * '.r +lk .i J 

 where real end imaginary parts of both waveniuabers are both positive 

 and wuere the Imaginary parts are stall (and in the end vanishingly small 

 compared with the real parts. 



The rationale Cor this is as follow*. We start by assuming a titse 

 factor exp(-lut) vith u > 0. (Ua could just as veil take exp(+iwt), but 

 the choice of exp(-lut) is helpful for reasons that should eserge later.) 

 Then mi 4 + » the phase factor of an outgoing wave mat be exp(+ik x) 

 where k » w/C . Giving k a small positive imaginary part therefore 

 oakeo an outgoing wave decay as x ■* + «, like exp(-k .x), corresponding 

 to the presence of small isceraal dissipation in the string. Similarly, 

 mx* -*ea outgoing wave will have the p! *»s?.e factor expO-ik k), and 

 then will also be exponentially dasped if we give fc a small positive 

 imaginary part. 



We therefore now know that for x > 0, y(x) is a continuous function 

 which decays like exp(-k sl x) as x * ». Ita HALF-RAKGE FCKI&IER TIUKSFORM 

 then has certain properties as a function of coisplex wavenusber s (that 

 being a useful syabol for the U9veaus.be r which does not have any particu- 

 lar significance, the symbol k for exasple often being associated with 

 a particular vsvenuaber). Define 



v> - { 



y(*) esp lex da. (2.3) 



