spherical%20polar%20coordinates
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11: 10.47 Definitions and Basic Properties
12: 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
… ►13: 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).
14: 10.56 Generating Functions
15: 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
…16: 22.18 Mathematical Applications
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►In polar coordinates, , , the lemniscate is given by , .
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►Discussion of parametrization of the angles of spherical trigonometry in terms of Jacobian elliptic functions is given in Greenhill (1959, p. 131) and Lawden (1989, §4.4).
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17: 10.51 Recurrence Relations and Derivatives
18: 29.18 Mathematical Applications
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§29.18(i) Sphero-Conal Coordinates
… ►(29.18.5) is the differential equation of spherical Bessel functions (§10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1). ►§29.18(ii) Ellipsoidal Coordinates
►The wave equation (29.18.1), when transformed to ellipsoidal coordinates : … ►§29.18(iii) Spherical and Ellipsoidal Harmonics
…19: 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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