# §10.68 Modulus and Phase Functions

## §10.68(i) Definitions

10.68.2

where , , , and are continuous real functions of and , with the branches of and chosen to satisfy (10.68.18) and (10.68.21) as . (See also §10.68(iv).)

## §10.68(ii) Basic Properties

10.68.7

With arguments suppressed,

Equations (10.68.8)–(10.68.14) also hold with the symbols , , , and replaced throughout by , , , and , respectively. In place of (10.68.7),

## §10.68(iii) Asymptotic Expansions for Large Argument

When is fixed, , and

## §10.68(iv) Further Properties

Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). However, care needs to be exercised with the branches of the phases. Thus this reference gives (Eq. (6.10)), and (Eqs. (10.20) and (Eqs. (10.26b)). However, numerical tabulations show that if the second of these equations applies and is continuous, then ; compare Abramowitz and Stegun (1964, p. 433).