which can be rearranged as follows. We have (s + y ) ( s ~ Y ) " k 2 » 



's 's o 



hence 



ft - -i £n . 



s + Y 



■"Hfa 



m 



(10.15) 



The definitions (10.14) and (10.15) of the function cos -1 (s/k ) 



are not those given in most books on mathematical functions. In those 



books the square root function is usually understood to have a branch 



cut from -k to +k , and is quite different from the function v which 

 o o s 



occurs in wave applications. In particular, if the cut gees from -k 

 JL ° 



to +k , (s 2 - k 2 ) 2 behaves like s both as s + + » and 8 -*■ - <*>, whereas 

 o ° 



Y behaves like s as s •* + °° but as - s when s ■* - °°. 

 a 



With the definition 



cos -1 (—. = + i in 



(^) 



(10.16) 



Consider the functions R(s), P_(s) defined in (10.11), In the first 



place, no new branch cuts are introduced by the logarithm. A branch 



cut would be needed only if s + Y ° were possible for some value of s, 



and no such value of s exists. Thus the only singularities are the branch 



points at s ■ ± k . Consider the function P. (s) near the point s ■ + k , 

 o + o 



writing s • k + u where u Is small. We have 



iu 2 (2k + u) 2 



?. (k + u) 

 + o 



£n . 



J. 1^ 



u 2 (2k + u) 2 



i J 2 (2k ) 2 u 2 (2k )' 



+ 0(u 2 ) 



(10.17) 



77 



