.•-,■•"- s 



• 5 i>:«»w!*» 



as u + 0, so that P (s) is In fact single-valued near s - k . The branch 



point singularity of the cos -1 cancels that of Y at s • k , and hence 



s o 



P (s) is analytic in R . Similarly P (s) is analytic in R_, the branch 



point singularity of y at s ■ - k being cancelled by that of 



cos -1 (- s/k ) there, 

 o 



To verify (10.12) we have, for s in D, 



».r. 



P + (S) + P_(6) 



IY S /s + Y\ /" s + Y s \ 



iY. 



*n(- 1) 



since with (10.13) fcn(-l) - - iff. 



To find the behavior at infinity we note that as |s| •*■ » in R 



Y c - s 



2s 2 " 8 



(10.18) 



if the approach to infinity is below the branch cut from s ■ + k^, 

 k 2 



's 



1 o + 

 IP 8 8 h 



(10.19) 



if s goes to infinity above the cut. Since P (s) has no singularity at 



8 ■ + k it does not matter which of these is used, and we find 

 o 



»♦«-*(■-§♦••) h(r) -%*•"] 



.Jstn (&\ + 0(s _1 In s) 



(10.20) 



These properties of the P + functions arise in a great many applications 

 to wave problems. Corresponding results for the convected wave square 



78 



