Airy Functions of Complex Argument
A 0 z = A i z ) A i l ) = A i ( ~ e ~ ' ~ ) , Ai,,(z) = Ai(ze2"'l"). (8.02) We also denote the sectors Iph zl< +K, +a < phz < n, and -n < phz < -+n by So, S,, and S-,, respectively; see Fig. 8.1. And in considering the Ai,(z) and Sj, and also auxiliary functions introduced in g8.3 below, we enumerate the s u f i j modulo 3; thus
From Chapter 4, $4 and Chapter 1, g8.3, we know that the asymptotic expansions ( 1.07), (1 .OR), and (1.09) are valid with x replaced by z and < denoting the principal value of 3z3I2, provided that I ph z( < K - 8 in the case of (1.07), and I ph z) < 3n-6 in the case of (1.08) and (1.09). Accordingly, at injnity Ai(z) is recessive within So, and dominant within S, and S-, . From this it follows that Aij(z) is recessive within
Sj and dominant within Sj-I and Sj+,. Hence Ai,(z) and Aij+,(z) comprise a numerically satisfactory pair of solutions of (8.01) in Sj v Sj+,, but not within Sj-'.