504 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



where 



h' = (2/pY\m + ip'/2)i"\ ^^^-^^^ 



i = exp (^7^/2), m = w + 1. 



In (13.14) hi"^ = -h and in (13.21) bT'" = -h' since Ai(a) is a 

 single-valued function of a. It is interesting to note that the factor 

 i^^^ in the expression for U„{i^'^p) gives the direction of that one of the 

 three paths of steepest descent (in the ^plane) which is not traversed 

 in getting Un{t'^p). The same sort of thing is true for the remaining 

 expressions in (13.14) and (13.21). 



The functions 



'Un(z) = exp (z'/2)d[Un(z) exp (- z'/2)]/dz, 



defined by (4.19), may be computed from (13.21) when z = C^'^p. We 

 need the relations d/dz = I'^d/dp and 



m = -tpV2 + h'{p''/2if\ 



(13.23) 

 dV/dp = i2/Z)(2ipy'\i - m/p) = i(2ip)"' - 25V3p, 



which follow from the definition of 6'. When the differentiations are 

 carried out we obtain 



'UniiT'^p) - (2py''C'r"'Ai'(b'i'-"'), 



'VniiT'^p) ^ i2py"C'i"'Ai'(h'i''"), (13.24) 



'Wr.{t~"'p) ^ {2pf"C'i'"'Ai'{h'i-^"), 



In these expressions the prime on the Airy integral denotes its deriva- 

 tive: 



Ai'{a) = dAi{a)/da. (13.25) 



