88 BELL SYSTEM TECHNICAL JOURNAL 



K into two parts (10). The first part connects <x + m to Wi and the second 

 part connects Wi to oo — i-jr. The values of iv on these two parts will be 

 denoted by Wj and W//, respectively. Corresponding to each value of r there 

 is a value Wi and a value wu (in fact it turns out that Wn is the conjugate 

 complex of Wj). Changing the variable of integration in (Al-23) from iv to 

 r, and remembering that K starts at oo + tV, gives 



-PCvo, u) = -^ — ^-r^ / e \ wi -— wjj dr (Al-24) 



liri Jo [_aT dr J 



Since m is large, most of the contribution to the value of the integral 

 comes from around r = or w = wi. In order to obtain an expression for 

 the integrand in this region we note that, because F'(vi) = 0, the Taylor 

 series for (Al-22) is of the form 



T = —b'ziw — wi)- — bsiw — wiY — bi(w — WiY — • • • (Al-25) 



The circle of convergence of this series is centered on ic<i and extends out to 

 w = dziir, these points being the nearest singularities of F(l + c"^') as may 

 be seen by setting t) = 1 + c"' in (Al-11) and observing that the singulariiies 

 of log V — t/v in the finite portion of the w plane occur at odd multiples of 

 dziir. We imagine the branch cuts associated with log v to run out to the 

 right from these points along lines parallel to the real w axis. Since (Al-25) 

 has a non-zero radius of convergence, the same is true of the two series ob- 

 tained from it by inversion, namely 



■T-l/2 1/2 , , /T,2 

 Wi — Wl = 102 T + OzT/ 102 



+ i[b7% - 5b7'bl/4y/2bl" + • • • 



and the series for Wn — wi obtained from (Al-26) by changing the sign of 

 i. Differentiation of these two series gives a series for d{u'i — Wu)/dT which 

 also converges for sufficiently small | r | (putting aside the term in t~^'-), 

 and which, when put in (Al-24), leads to 



That this is an asymptotic expansion holding for large values of m follows 

 from a lemma given by Watson (11). The conditions of the lemma hold 

 since we have already shown that the series for d{ivi — Wii)/dT converges 

 for I T I small enough. Furthermore, d{wi — iVii)/dT is bounded for c ^ t 

 where t is real and < a ^ the radius of convergence of (Al-26). This 

 follows the fact that 



*" [3 ' = '-'■"(! + ^">"i" 



di 



