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11: 33.14 Definitions and Basic Properties
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►Again, there is a regular singularity at with indices and , and an irregular singularity of rank 1 at .
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►An alternative formula for is
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►When and the quantity may be negative, causing and to become imaginary.
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►Note that the functions , , do not form a complete orthonormal system.
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►With arguments suppressed,
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12: 33.8 Continued Fractions
13: 33.9 Expansions in Series of Bessel Functions
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►where the function is as in §10.47(ii), , , and
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33.9.3
,
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►Here , , and
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33.9.5
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►For other asymptotic expansions of see Fröberg (1955, §8) and Humblet (1985).
14: 33.16 Connection Formulas
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§33.16(i) and in Terms of and
… ►where is given by (33.2.5) or (33.2.6). … ►and again define by (33.14.11) or (33.14.12). … ►and again define by (33.14.11) or (33.14.12). … ►When denote , , and by (33.16.8) and (33.16.9). …15: 33.2 Definitions and Basic Properties
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►This differential equation has a regular singularity at with indices and , and an irregular singularity of rank 1 at (§§2.7(i), 2.7(ii)).
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►The normalizing constant
is always positive, and has the alternative form
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is the Coulomb phase shift.
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and are complex conjugates, and their real and imaginary parts are given by
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►As in the case of , the solutions and are analytic functions of when .
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16: 33.19 Power-Series Expansions in
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►Here is defined by (33.14.6), is defined by (33.14.11) or (33.14.12), , , and
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►with , and
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17: 33.20 Expansions for Small
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►where
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►As with and fixed,
…where is given by (33.14.11), (33.14.12), and
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►For a comprehensive collection of asymptotic expansions that cover and as and are uniform in , including unbounded values, see Curtis (1964a, §7).
These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders and .