76 BELL SYSTEM TECHNICAL JOURNAL 



where a and 13 are new variables whose absolute values never exceed 

 (2 g log qY'- in the restricted region of integration of (5-6). From (4-11) 



p.iu'x.) dn dx, = iL±-^ (^^^"'' Uz''')e-'^^''''^^"'-'' da d& (5-12) 



in which 



s = ux, = 4r(l -f r){q + a)iq + fi) (5-13) 



In Appendix II it is shown that 



/ (^^''-) = <l'^"'e-'z^''exp[(q' + zr--\-V] 



r{q + i)(g2 + zyi^iq + (^2 + 2)i/2]« ^^■''*^ 



where | P' | < 1/(2^ — 1) when ^ > 1. Upon using (5-10) and (5-14) the right 

 hand side of (5-12) may be written as 



da di3{27ry"'a + r)(2r)-''((/2 -f z)-'" exp [- (1 + r)(2q + a i- /3) (5-15) 



+ /(s) - log T{q + 1) + 0(l/q)] 



with 



f(z) = q\ogz- q log [q + (^^ -f zY''] + (^^ + s)!/^ (5-16) 



The value Zo of z corresponding to the central point («2, -Vj) of />o(«, -Vo) is 

 obtained by putting a = /3 = in (5-13): 



So = 4^1 + 0?- 



(5-17) 



2 - So = 4K1 + r)[5(« + iS) + ai3]. 



Since we are interested in the form of Pq{u, Xq) in the restricted region of 

 integration of (5-6) we expand /(s) about s = Z2 in a Taylor's series plus a 

 remainder term. 



/(s) = q log 2rq -f (y(l -f 2r) + {z - z^M{Arq) 



(5-18) 



(2 - z,f (z - 2,)^ r (^3 + g)^(3^3 - y) - 



32rY(l -f 2r) "^ 3! [ 8 s' ^3 



In the last term S3 = S2 + (2 — 20)6, 0^6^ 1, ^3 = </'■+ -3. The work of 

 obtaining this expansion is simplified if (q- -\- s)''- is replaced by ^ in (5-16) 

 before differentiating. For example, by using 2^'| = 1, it can be shown that 

 f'(z) is simply (q + |)/(2s). When the extreme values of a and f3 are put in 

 (5-17), it is seen that s — Zo does not exceed 0{(f'- log^'- q) in the restricted 

 region of integration. In the last term of (5-18) Z3 is 0(5^), ^3 is O(^) and con- 

 sequently the last term itself is 0(</~^/- log^'- q). 



