For the modulus functions and see §12.14(x).
the branch of being zero when and defined by continuity elsewhere.
Here and are the even and odd solutions of (12.2.3):
where and satisfy the recursion relations
For the notation see §10.2(ii). When
For the notation see §13.2(i).
with given by (12.14.7). Then as
The coefficients and are obtainable by equating real and imaginary parts in
The differential equation
follows from (12.2.3), and has solutions . For real and oscillations occur outside the -interval . Airy-type uniform asymptotic expansions can be used to include either one of the turning points . In the following expansions, obtained from Olver (1959), is large and positive, and is again an arbitrary small positive constant.
uniformly for , with , , , and as in §12.10(vii). For the corresponding expansions for the derivatives see Olver (1959).
As noted in §12.14(ix), when is negative the solutions of (12.2.3), with replaced by , are oscillatory on the whole real line; also, when is positive there is a central interval in which the solutions are exponential in character. In the oscillatory intervals we write
where is defined in (12.14.5), and (0), , (0), and are real. or is the modulus and or is the corresponding phase. Compare §12.2(vi).
For properties of the modulus and phase functions, including differential equations and asymptotic expansions for large , see Miller (1955, pp. 87–88). For graphs of the modulus functions see §12.14(iii).
For asymptotic expansions of the zeros of and , see Olver (1959).