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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



(10.2) and subtracting the corresponding expression for zUn{z) we 

 obtain 



'Un{z) = U'n{z) -Z Un(z) 



~ {z^ — 2riif''^ [{U contribution) — {t\ contribution) (12.2) 



— ihe''"' contribution)] 



where 'Un(z) is the function defined by (4.19). The same is true for 

 'Vn(z) and 'Wn(z). 



Consideration of the various paths of integration shown in Section 11 

 leads to the results shown in Table 12.1. The leading terms of the 

 asymptotic expansions are listed for the various regions of the m-plane 



Table 12.1 — Leading Terms in the Asymptotic Expansions for 



Un{z), Vn(z), W„{z) WHEN Z = I'^p, p > 



shown in Fig. 11.2. If the next order terms are required, they may be 

 obtained from (10.4) and (10.5). 



The notation used in Table 12.1 is as follows: 



z = i^'p, m = n -\- 1, i — exp (iV/2), 



- 7r/2 < arg yyi ^ 3x/2, - 7r/4 < arg ^o ^ 37r/4, 



- x/2 < arg Up- - 2m) ^ 37r/2, - 37r/4 < arg ^i S 57r/4, 



t, = [(''■' p + dp' - 2?ny'']/2, h = [t"p - Up' - 2mf']/2, 



Ao = [dip' - 2m)-"y2i7r"'] exp/(/o), ■ (i2.3) 



Ar = [tr-(ip' - 2mr'''/2r"'] expfiU), 



Hio) = zto-i- - - m log /o = 2(1 - log -^ - log j) + ^ P'c, 

 f{h) = zti + - - m log /i = _M - log - - log - ) + I ph. 



