# §9.8 Modulus and Phase

## §9.8(i) Definitions

Throughout this section is real and nonpositive.

Graphs of and are included in §9.3(i). The branches of and are continuous and fixed by . (These definitions of and differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).)

## §9.8(iii) Monotonicity

As increases from to 0 each of the functions , , , , , is increasing, and each of the functions , , is decreasing.

## §9.8(iv) Asymptotic Expansions

In (9.8.20) and (9.8.21) the remainder after terms does not exceed the th term in absolute value and is of the same sign, provided that for (9.8.20) and for (9.8.21).

For higher terms in (9.8.22) and (9.8.23) see Fabijonas et al. (2004). Also, approximate values (25S) of the coefficients of the powers , , are available in Sherry (1959).