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11: 33.18 Limiting Forms for Large
12: 18.18 Sums
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Expansion of functions
►In all three cases of Jacobi, Laguerre and Hermite, if is on the corresponding interval with respect to the corresponding weight function and if are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in sense. … ►
18.18.12
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18.18.18
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18.18.19
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13: 33.1 Special Notation
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►The main functions treated in this chapter are first the Coulomb radial functions , , (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions , , , (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions.
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Curtis (1964a):
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Greene et al. (1979):
nonnegative integers. | |
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, .
, , .
14: 33.13 Complex Variable and Parameters
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►The functions , , and may be extended to noninteger values of by generalizing , and supplementing (33.6.5) by a formula derived from (33.2.8) with expanded via (13.2.42).
►These functions may also be continued analytically to complex values of , , and .
The quantities , , and , given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that
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33.13.1
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33.13.2
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15: 24.16 Generalizations
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►For , Bernoulli and Euler polynomials of order
are defined respectively by
…When they reduce to the Bernoulli and Euler numbers of
order
:
…Also for ,
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►For extensions of to complex values of , , and , and also for uniform asymptotic expansions for large and large , see Temme (1995b) and López and Temme (1999b, 2010b).
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►(This notation is consistent with (24.16.3) when .)
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16: 33.15 Graphics
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§33.15(i) Line Graphs of the Coulomb Functions and
► ► ► … ►§33.15(ii) Surfaces of the Coulomb Functions , , , and
…17: DLMF Project News
error generating summary18: 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).
19: 16.17 Definition
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►where the integration path separates the poles of the factors from those of the factors .
There are three possible choices for , illustrated in Figure 16.17.1 in the case , :
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(i)
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(ii)
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(iii)
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goes from to . The integral converges if and .
is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all () if , and for if .
is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all if , and for if .