Coulomb radial functions
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1—10 of 22 matching pages
1: 33.3 Graphics
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§33.3(i) Line Graphs of the Coulomb Radial Functions and
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33.3.1
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ⓘ
- Defines:
- : envelope of Coulomb functions
- Symbols:
- : irregular Coulomb radial function, : irregular Coulomb radial functions, : regular Coulomb radial function, : nonnegative integer, : nonnegative real variable and : real parameter
- Permalink:
- http://dlmf.nist.gov/33.3.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.3(i), §33.3 and Ch.33
§33.3(ii) Surfaces of the Coulomb Radial Functions and
…2: 33.5 Limiting Forms for Small , Small , or Large
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33.5.6
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- Symbols:
- : normalizing constant for Coulomb radial functions, : double factorial, : factorial (as in ) and : nonnegative integer
- Permalink:
- http://dlmf.nist.gov/33.5.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.5(ii), §33.5 and Ch.33
33.5.9
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- Symbols:
- : normalizing constant for Coulomb radial functions, : asymptotic equality, : the ratio of the circumference of a circle to its diameter, : double factorial, : base of natural logarithm, : nonnegative integer and : real parameter
- Referenced by:
- §33.23(iv), §33.5(iv)
- Permalink:
- http://dlmf.nist.gov/33.5.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.5(iv), §33.5 and Ch.33
3: 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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Greene et al. (1979):
, , .
4: 33.2 Definitions and Basic Properties
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33.2.2
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- Defines:
- : outer turning point for Coulomb radial functions
- Symbols:
- : nonnegative integer and : real parameter
- Referenced by:
- §33.12(i), §33.14(i)
- Permalink:
- http://dlmf.nist.gov/33.2.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.2(i), §33.2 and Ch.33
33.2.3
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ⓘ
- Defines:
- : regular Coulomb radial function
- Symbols:
- : normalizing constant for Coulomb radial functions, : Whittaker confluent hypergeometric function, : imaginary unit, : nonnegative integer, : nonnegative real variable and : real parameter
- Referenced by:
- §13.18(vi), §33.2(ii)
- Permalink:
- http://dlmf.nist.gov/33.2.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.2(ii), §33.2 and Ch.33
33.2.5
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- Defines:
- : normalizing constant for Coulomb radial functions
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : factorial (as in ), : imaginary unit, : nonnegative integer and : real parameter
- A&S Ref:
- 14.1.7
- Referenced by:
- §33.10(ii), §33.10(iii), §33.16(i)
- Permalink:
- http://dlmf.nist.gov/33.2.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.2(ii), §33.2 and Ch.33
33.2.12
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- Symbols:
- : Wronskian, : irregular Coulomb radial function, : irregular Coulomb radial functions, : regular Coulomb radial function and : nonnegative integer
- A&S Ref:
- 14.2.4
- Referenced by:
- §33.23(v)
- Permalink:
- http://dlmf.nist.gov/33.2.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.2(iv), §33.2 and Ch.33
33.2.13
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- Symbols:
- : irregular Coulomb radial function, : regular Coulomb radial function, : nonnegative integer and : real parameter
- A&S Ref:
- 14.2.5
- Permalink:
- http://dlmf.nist.gov/33.2.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.2(iv), §33.2 and Ch.33
5: 33.10 Limiting Forms for Large or Large
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33.10.2
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- Symbols:
- : phase of Coulomb functions, : asymptotic equality, : exponential function, : imaginary unit, : irregular Coulomb radial functions, : nonnegative integer, : nonnegative real variable and : real parameter
- A&S Ref:
- 14.6.5
- Permalink:
- http://dlmf.nist.gov/33.10.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.10(i), §33.10 and Ch.33
6: 33.13 Complex Variable and Parameters
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33.13.1
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- Symbols:
- : normalizing constant for Coulomb radial functions, : Coulomb phase shift, : gamma function, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : nonnegative integer and : real parameter
- Permalink:
- http://dlmf.nist.gov/33.13.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.13 and Ch.33
33.13.2
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- Symbols:
- : normalizing constant for Coulomb radial functions, : nonnegative integer, : real parameter and : factor
- Permalink:
- http://dlmf.nist.gov/33.13.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.13 and Ch.33
7: 33.8 Continued Fractions
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33.8.1
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- Symbols:
- : regular Coulomb radial function, : nonnegative integer, : factor, : factor and : factor
- Referenced by:
- §33.8, §33.8
- Permalink:
- http://dlmf.nist.gov/33.8.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.8 and Ch.33
33.8.2
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- Symbols:
- : imaginary unit, : irregular Coulomb radial functions, : nonnegative integer, : nonnegative real variable, : real parameter, : coefficient, : coefficient and : coefficient
- Referenced by:
- §33.8
- Permalink:
- http://dlmf.nist.gov/33.8.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.8 and Ch.33
8: 33.6 Power-Series Expansions in
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33.6.1
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- Symbols:
- : normalizing constant for Coulomb radial functions, : regular Coulomb radial function, : nonnegative integer, : nonnegative integer, : nonnegative real variable, : real parameter and : coefficient
- A&S Ref:
- 14.1.4, 14.1.5
- Referenced by:
- §33.6
- Permalink:
- http://dlmf.nist.gov/33.6.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.6 and Ch.33
33.6.2
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ⓘ
- Symbols:
- : normalizing constant for Coulomb radial functions, : regular Coulomb radial function, : nonnegative integer, : nonnegative integer, : nonnegative real variable, : real parameter and : coefficient
- A&S Ref:
- 14.1.4, 14.1.5
- Referenced by:
- §33.6
- Permalink:
- http://dlmf.nist.gov/33.6.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.6 and Ch.33
33.6.5
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- Symbols:
- : phase of Coulomb functions, : gamma function, : Pochhammer’s symbol (or shifted factorial), : psi (or digamma) function, : base of natural logarithm, : factorial (as in ), : imaginary unit, : irregular Coulomb radial functions, : principal branch of logarithm function, : nonnegative integer, : nonnegative integer, : nonnegative real variable, : real parameter and : variable
- A&S Ref:
- 14.1.14 (equivalent)
- Referenced by:
- §33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5)
- Permalink:
- http://dlmf.nist.gov/33.6.E5
- Encodings:
- pMML, png, TeX
- Errata (effective with 1.0.19):
-
On the right-hand side the factor in the denominator was
incorrectly written as .
This has been corrected to
.
Reported 2018-05-15 by Ian Thompson
- See also:
- Annotations for §33.6 and Ch.33
9: 33.7 Integral Representations
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33.7.1
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- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : imaginary unit, : integral, : regular Coulomb radial function, : nonnegative integer, : nonnegative real variable and : real parameter
- Referenced by:
- §33.7, §33.7
- Permalink:
- http://dlmf.nist.gov/33.7.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.7 and Ch.33
33.7.2
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ⓘ
- Symbols:
- : normalizing constant for Coulomb radial functions, : differential of , : base of natural logarithm, : factorial (as in ), : imaginary unit, : integral, : irregular Coulomb radial functions, : nonnegative integer, : nonnegative real variable and : real parameter
- A&S Ref:
- 14.3.1 (modified)
- Referenced by:
- §33.7
- Permalink:
- http://dlmf.nist.gov/33.7.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.7 and Ch.33
33.7.3
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ⓘ
- Symbols:
- : normalizing constant for Coulomb radial functions, : the ratio of the circumference of a circle to its diameter, : differential of , : exponential function, : base of natural logarithm, : factorial (as in ), : hyperbolic cosine function, : hyperbolic tangent function, : imaginary unit, : integral, : arctangent function, : irregular Coulomb radial functions, : nonnegative integer, : nonnegative real variable and : real parameter
- A&S Ref:
- 14.3.3
- Referenced by:
- §33.7
- Permalink:
- http://dlmf.nist.gov/33.7.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.7 and Ch.33
33.7.4
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- Symbols:
- : normalizing constant for Coulomb radial functions, : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : factorial (as in ), : imaginary unit, : integral, : irregular Coulomb radial functions, : nonnegative integer, : nonnegative real variable and : real parameter
- A&S Ref:
- 14.3.2
- Referenced by:
- §33.7, §33.7
- Permalink:
- http://dlmf.nist.gov/33.7.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §33.7 and Ch.33
10: 33.11 Asymptotic Expansions for Large
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33.11.1
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- Symbols:
- : phase of Coulomb functions, : Pochhammer’s symbol (or shifted factorial), : Poincaré asymptotic expansion, : base of natural logarithm, : factorial (as in ), : imaginary unit, : irregular Coulomb radial functions, : nonnegative integer, : nonnegative integer, : nonnegative real variable, : real parameter, : coefficient and : coefficient
- A&S Ref:
- 14.5.9
- Referenced by:
- Erratum (V1.0.19) for Equation (33.11.1), Erratum (V1.0.22) for Equation (33.11.1)
- Permalink:
- http://dlmf.nist.gov/33.11.E1
- Encodings:
- pMML, png, TeX
- Errata (effective with 1.0.22):
-
Previously this formula was expressed as an equality. Since this formula
expresses an asymptotic expansion, it has been corrected by using instead
an relation.
Reported 2019-01-29 by Gergő Nemes
- Errata (effective with 1.0.19):
-
Originally the factor in the denominator on the right-hand side
was written incorrectly as .
This has been corrected to .
Reported 2018-05-15 by Ian Thompson
- See also:
- Annotations for §33.11 and Ch.33
33.11.4
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- Symbols:
- : phase of Coulomb functions, : base of natural logarithm, : imaginary unit, : irregular Coulomb radial functions, : nonnegative integer, : nonnegative real variable, : real parameter, : function and : function
- Permalink:
- http://dlmf.nist.gov/33.11.E4
- Encodings:
- pMML, png, TeX
- Clarification (effective with 1.0.22):
- The arguments of some of the functions in these equations were included to improve clarity of the presentation.
- See also:
- Annotations for §33.11 and Ch.33