Hankel
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1: 10.1 Special Notation
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►The main functions treated in this chapter are the Bessel functions , ; Hankel functions , ; modified Bessel functions , ; spherical Bessel functions , , , ; modified spherical Bessel functions , , ; Kelvin functions , , , .
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►Abramowitz and Stegun (1964): , , , , for , , , , respectively, when .
►Jeffreys and Jeffreys (1956): for , for , for .
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►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
2: 10.5 Wronskians and Cross-Products
3: 10.11 Analytic Continuation
4: 10.4 Connection Formulas
5: 10.2 Definitions
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Bessel Functions of the Third Kind (Hankel Functions)
►These solutions of (10.2.1) are denoted by and , and their defining properties are given by … ►The principal branches of and are two-valued and discontinuous on the cut . … ►Branch Conventions
…6: 10.47 Definitions and Basic Properties
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10.47.5
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and are the spherical Bessel
functions of the first and second kinds, respectively; and are the spherical Bessel functions of the
third kind.
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►Many properties of , , , , , , and follow straightforwardly from the above definitions and results given in preceding sections of this chapter.
For example, , , , , , , and are all entire functions of .
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10.47.15
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7: 10.16 Relations to Other Functions
8: 9.6 Relations to Other Functions
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§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions
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9.6.6
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9.6.7
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9.6.8
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§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions
…9: 10.7 Limiting Forms
§10.7 Limiting Forms
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10.7.2
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►For and when combine (10.4.6) and (10.7.7).
For and when and combine (10.4.3), (10.7.3), and (10.7.6).
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►For the corresponding results for and see (10.2.5) and (10.2.6).