1018 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



at a real negative point 



s = —a > —X. 

 For large L, the asymptotic behavior is given by 



-aL 



r, /T\ Xe "^ r aL 



(X - a)[l + 5(X - a)]N<. Ll + S(X - a). 



N 



where the error term is 0(L^~^ e~"^) if A" ^ 1. Such a formula, then, is 

 a good approximation for fixed N as L increases; for fixed L, however, 

 it will fail to be good for sufficiently large A^. 



If iV = 0, the asymptotic form is 



Fo(L) ^ 7- rr- j -TT r-, 6 , 



(X — a)[l + 5(X — a) J 



but the error term now decreases at a more rapid rate, as may be seen 

 by including the contributions of some of the complex poles of /o(s). To 

 find these poles, set 



s + X = Xe . 



If 



s = —X + r exp (id), 



one obtains the simultaneous real system 



2irm — 6 = 8r sin 6 (m integer), 



log (r/X) = —6r cos 6. 



The first equation defines an infinite family of curves in the s-plane (see ! 

 Fig. 5). The second equation defines a single curve which intersects the 

 family at poles of p{s). 



2.5 Bounds on Fo(L) 



As in Part I, we may derive bounds on Fo{L) from the integral equa- , 

 tion, and obtain ] 



(l --^ e^^'-e-^"-^^' g Fo(L) ^ e'^'^e-^''-^^'. 



Since a = X^5 + 0(5") for small 5, the bounds are very close if X5 is not 

 too large. 



