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11: 28.23 Expansions in Series of Bessel Functions
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28.23.6
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28.23.8
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28.23.10
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28.23.12
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►When the series in the even-numbered equations converge for and , and the series in the odd-numbered equations converge for and .
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12: 33.6 Power-Series Expansions in
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33.6.1
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►where , , and
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33.6.3
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►where and (§5.2(i)).
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►Corresponding expansions for can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).
13: 33.5 Limiting Forms for Small , Small , or Large
§33.5 Limiting Forms for Small , Small , or Large
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33.5.6
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§33.5(iv) Large
►As with and () fixed, …14: 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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►The functions and are defined by
…An alternative formula for is
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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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15: 26.2 Basic Definitions
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►Given a finite set with permutation , a cycle is an ordered equivalence class of elements of where is equivalent to if there exists an such that , where and is the composition of with .
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►The total number of partitions of is denoted by .
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16: 33.8 Continued Fractions
17: 10.42 Zeros
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►For example, if is real, then the zeros of are all complex unless for some positive integer , in which event has two real zeros.
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has no zeros in the sector ; this result remains true when is replaced by any real number
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For the number of zeros of in the sector , when is real, see Watson (1944, pp. 511–513).
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18: 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). …19: 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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