With arguments suppressed,
When is fixed, , and
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).