500 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



for the electromagnetic field which contain Airy integrals may be re- 

 placed by more accurate, but also more complicated, expressions. In 

 dealing with our functions we shall work with the integrals and our 

 procedure is somewhat similar to that used by Rydbeck.* First however, 

 we point out that when we WTite (as suggested by the work of Cherry 

 and Tricomi) 



y = e-''"Tn{z), ax = z-{2n-\- lf\ 2(2n -f if "a = 1, (13.1) 



the differential equation (9.7) for the parabolic cylinder functions goes 

 into 



Pi-xy = 2-''\2n + lV"x\j. (13.2) 



ax- 



The Airy integrals Ai{x) and Bi{:x) (and also Ai[x exp (± f27r/3)]) 

 discussed later in this section are solutions of 



% - ^y = 0. (13.3) 



and therefore we expect that approximate solutions of (13.2) are given 

 by, for example, 



y = CiAiixOil + 0(rr~")] (13.4) 



where the 0(rr^'^) term corresponds to the particular integral of (13.2) 

 when the y on the right hand side is replaced by its approximate value 

 Aiixi"). Here Ci is independent of x (or z) but may depend on n, and v 

 may be or ±4/3. 



Since the labor of computing Ci is considerable, we shall work out the 

 approximations directly from the integrals. 



We shall consider the case z = i^''p, p > 0, first. When n + 1 = m 

 = mo ^ ip^/2 the saddle points to and ti coincide at ti = i^'^p/2. Con- 

 sequently only those portions of the paths of steepest descent which 

 lie near ^2 are of importance. This is true even if m is not exactly equal 

 to mo. We therefore regard 



f{t) = -t' -{- 2zt - m log t (13.5) 



in (10.1) as a function of the two variables t and m (linear in m) with z 

 fixed at i^'^p. Expanding (13.5) about t ^ t-i , ni = m^ gives 



^ '"wio Wo , mo (m — mo) , viq 



— — — log — - — log 



2 2^2 2 ^ 2 J (13.6) 



- 4(/ - ttf/^z - 2(m - mo){t - /o)/^ + • • • 



/(o = I + 



* Page 87 of Reference 21 cited on page 478. 



