ABSTRACT
The ( map of a suitable progressive path linking a, with a2 is shown in Fig. 12.2. When the radius R of the semicircular arc tends to infinity, the variation of F vanishes. Therefore, from (1 1.14), &,(a2) = 0. Similarly both e2(a3) and e3(al) are zero. Accordingly, (1 1.13) reduces to
w1(z) + e-2xi13~2 (z) + e2x113w3 (z) = 0. (12.01) By comparing (11.07) with recessive properties of the Airy functions given in Chapter 1 I , 81 .1 and 8.1, we identify w,(z) = 2n1I2 Ai(z), w2 (r) = 2n1I2 Ai(~e-~" ' l~) , w,(z) = 2n1I2 Ai(ze2"'I3 1; accordingly, equation (12.01) agrees with (8.06) of Chapter 2. The point of the example is that the present theory yields the connection formula without recourse to integral representations.
