§9.8 Modulus and Phase
Contents
- §9.8(i) Definitions
- §9.8(ii) Identities
- §9.8(iii) Monotonicity
- §9.8(iv) Asymptotic Expansions
§9.8(i) Definitions
Throughout this section is real and nonpositive.
9.8.1 |
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9.8.2 |
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9.8.3 |
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9.8.4 |
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9.8.5 |
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9.8.6 |
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9.8.7 |
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9.8.8 |
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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).)
In terms of Bessel functions, and with ,
9.8.9 |
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9.8.10 |
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9.8.11 |
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9.8.12 |
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§9.8(ii) Identities
Primes denote differentiations with respect to , which is continued to be
assumed real and nonpositive.
9.8.13 |
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9.8.14 |
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9.8.15 |
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9.8.16 |
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9.8.17 |
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9.8.18 |
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9.8.19 |
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§9.8(iii) Monotonicity
As increases from to each of the functions
, , ,
, ,
is increasing, and each of the functions
, ,
is decreasing.
§9.8(iv) Asymptotic Expansions
As
9.8.20 |
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9.8.21 |
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9.8.22 |
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9.8.23 |
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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), (9.8.22) and (9.8.23), 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).
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