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11: 10.49 Explicit Formulas
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§10.49(i) Unmodified Functions
… ►§10.49(ii) Modified Functions
… ►§10.49(iii) Rayleigh’s Formulas
… ►§10.49(iv) Sums or Differences of Squares
… ►12: 10.53 Power Series
§10.53 Power Series
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10.53.3
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10.53.4
►For and combine (10.47.10), (10.53.1), and (10.53.2).
For combine (10.47.11), (10.53.3), and (10.53.4).
13: 18.39 Applications in the Physical Sciences
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►Now use spherical coordinates (1.5.16) with instead of , and assume the potential to be radial.
…By (1.5.17) the first term in (18.39.21), which is the quantum kinetic energy operator , can be written in spherical coordinates
as
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a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s
… ►c) Spherical Radial Coulomb Wave Functions
…14: 1.5 Calculus of Two or More Variables
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§1.5(ii) Coordinate Systems
… ►Polar Coordinates
… ►Cylindrical Coordinates
… ►Spherical Coordinates
… ►For applications and other coordinate systems see §§12.17, 14.19(i), 14.30(iv), 28.32, 29.18, 30.13, 30.14. …15: 10.56 Generating Functions
16: 10.60 Sums
§10.60 Sums
►§10.60(i) Addition Theorems
… ►§10.60(ii) Duplication Formulas
… ►For further sums of series of spherical Bessel functions, or modified spherical Bessel functions, see §6.10(ii), Luke (1969b, pp. 55–58), Vavreck and Thompson (1984), Harris (2000), and Rottbrand (2000). ►§10.60(iv) Compendia
…17: 10.51 Recurrence Relations and Derivatives
18: 10.57 Uniform Asymptotic Expansions for Large Order
§10.57 Uniform Asymptotic Expansions for Large Order
►Asymptotic expansions for , , , , , and as that are uniform with respect to can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). Subsequently, for the connection formula (10.47.11) is available. ►For the corresponding expansion for use ►
10.57.1
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19: 10.58 Zeros
§10.58 Zeros
►For the th positive zeros of , , , and are denoted by , , , and , respectively, except that for we count as the first zero of . … ►20: 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 , , , .
For the spherical Bessel functions and modified spherical Bessel functions the order is a nonnegative integer.
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►Abramowitz and Stegun (1964): , , , , for , , , , respectively, when .
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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).