476 THK BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



U-n-i(iz) = r^'V - t''")KWniz) = -(2i)-\'%-''Wn(z)/T(-n), 



U-n-i{-iz) = i'^-'ii"' - r'lKVniz), 



K = e-''T(n + l)/7r^'^2"+^ ^^'^^^ 



Equation (9.5) may be obtained from (9.1) by forming Tn(z) — 2zT'{z) 

 and integrating by parts. Equation (9.10) follows from (9.1) upon 

 joining the paths [/, F, W to obtain a closed path of integration. From 

 (9.11) it follows that when we have an expression for Vn{z) which holds 

 for all values of z and n, replacing i by — i (or i~^) gives the correspond- 

 ing expression for Wniz). Equations (9.13) may be obtained by using 

 the fact that U-n-iiiz) exp {z') etc. are solutions of the differential 

 equations (9.5) and (9.7). 



The relations (9.11) and (9.13) enable us to compute the values of 

 Un{z), Vniz), Wn{z) for z = %'' p aud z = r^''^ p and all n when the 

 values of any two are given for z = t^'^p and Re(n) ^ — 3^. 



It may be verified that as z becomes large and n remains fixed the 

 differential equation (9.5) has the asymptotic solutions 



z-^-'e'" (n±\ n+2 . , 



(9.14) 



where 



I arg z\ < TT and S2{ — n—l,iz) = i"Ksi(n,z), 



K being given by (9.13). In terms of these functions we have 



Vn{z) ^ —is2(n, z), < arg 2 < TT 



^ — Si(n, z) —is2(n, z), — 7r/2 < arg 2 < (9.15) 



~ -si(n, z) -i~*"^'^S2(n, z), -t < argz < -ir/2 



Wn{z) -^ is2(n, z), -TT < arg 2 < (9.16) 



Un{z) -^ si(n, z), -7r/2 < arg z < ir/2. (9.17) 



The first expression for Vn{z) in (9.15) follows when we note that the 

 leading term may be obtained from (9.1) by choosing the path of in- 

 tegration F to be f = 2 + T where r runs from — oo to -1- oo , and | z \ 

 is supposed to be large. (9.17) follows from the first of (9.15) and the 

 relation (9.13) between F_„_i(z2) and Un(z). Asymptotic expressions 

 for ir„(2) may be obtained by taking the conjugate complex of those 



