(s* - k!|) Y + (s) - y'"(0+) - isy" ((H) - s V (0+) + is s y((H) (6. A) 



(a* - k") Y (s) - -y m (0-) + isy" (0-) + s 2 y' (O-)-is'y(O-) (6.5) 

 o — 



the first of these holding in R , the second in R . Now the poles 

 s - +kj, s - +ikj lie in R + (see Figure 3), so that Y (s) will have 

 poles at +k t + ik unless we choose 



y'"(0+) -• ik x y"(0+) - k 2 y'(CH-) + ikj y(0+) - (6.6) 



y'"(0+) + k, y"(0+) + k 2 y'(0+) + kj y(0+) - (6.7) 



Similarly, Y (s) will have poles at s - -k , s - -ik in R unless 

 — o o — 



-y'"(0-) - ik y"(0-) + k 2 y'(O-) + ik 3 y(0-) - (6.8) 







-y'"(0-) + k y"(0-) - k 2 y'(O-) + k 3 y(O-) - (6.9) 







Whatever the boundary conditions at x - 0, conditions (6.6-6.9) must be 

 satisfied. (Two analogous relations could have been deduced in $2, but 

 it seeiLed unecessary co emphasize such a point at an early stage.) 



Four further conditions may be imposed at x ■ 0, at least one of 

 these being a nonhomogenous condition, so that we shall have a uniquely 

 soluble set of eight equations for the eight unknown boundary constants. 

 We shall not take any particular net of boundary conditions, as these 

 tend only to lead to complicated expressions without any special structure. 

 By addition of (6.4) and (6.5) we get 



(s- - k*) Y (s) + (s H - k") Y (s) - Q,(s) + Q (s) (6.10) 



27 



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