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NIST
33
Coulomb Functions
Variables
ρ
,
η
33.3
Graphics
33.5
Limiting Forms for Small
ρ
, Small
|
η
|
, or Large
ℓ
§33.4
Recurrence Relations and Derivatives
ⓘ
Keywords:
Coulomb functions: variables
ρ
,
η
,
derivatives
,
recurrence relations
Notes:
See
Powell (
1947
)
.
Referenced by:
§33.23(iv)
,
§33.23(iv)
Permalink:
http://dlmf.nist.gov/33.4
See also:
Annotations for
Ch.33
For
ℓ
=
1
,
2
,
3
,
…
, let
33.4.1
R
ℓ
=
1
+
η
2
ℓ
2
,
S
ℓ
=
ℓ
ρ
+
η
ℓ
,
T
ℓ
=
S
ℓ
+
S
ℓ
+
1
.
ⓘ
Defines:
R
ℓ
: factor (locally)
,
S
ℓ
: factor (locally)
and
T
ℓ
: factor (locally)
Symbols:
ℓ
: nonnegative integer
,
ρ
: nonnegative real variable
and
η
: real parameter
Referenced by:
§33.13
,
§33.8
Permalink:
http://dlmf.nist.gov/33.4.E1
Encodings:
TeX
,
TeX
,
TeX
,
pMML
,
pMML
,
pMML
,
png
,
png
,
png
See also:
Annotations for
§33.4
and
Ch.33
Then, with
X
ℓ
denoting any of
F
ℓ
(
η
,
ρ
)
,
G
ℓ
(
η
,
ρ
)
, or
H
ℓ
±
(
η
,
ρ
)
,
33.4.2
R
ℓ
X
ℓ
-
1
-
T
ℓ
X
ℓ
+
R
ℓ
+
1
X
ℓ
+
1
=
0
,
ℓ
≥
1
,
ⓘ
Symbols:
ℓ
: nonnegative integer
,
R
ℓ
: factor
,
T
ℓ
: factor
and
X
ℓ
: any Coulomb function
A&S Ref:
14.2.3
Permalink:
http://dlmf.nist.gov/33.4.E2
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§33.4
and
Ch.33
33.4.3
X
ℓ
′
=
R
ℓ
X
ℓ
-
1
-
S
ℓ
X
ℓ
,
ℓ
≥
1
,
ⓘ
Symbols:
ℓ
: nonnegative integer
,
R
ℓ
: factor
,
S
ℓ
: factor
and
X
ℓ
: any Coulomb function
A&S Ref:
14.2.1
Referenced by:
§33.6
Permalink:
http://dlmf.nist.gov/33.4.E3
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§33.4
and
Ch.33
33.4.4
X
ℓ
′
=
S
ℓ
+
1
X
ℓ
-
R
ℓ
+
1
X
ℓ
+
1
,
ℓ
≥
0
.
ⓘ
Symbols:
ℓ
: nonnegative integer
,
R
ℓ
: factor
,
S
ℓ
: factor
and
X
ℓ
: any Coulomb function
A&S Ref:
14.2.2
Referenced by:
§33.6
Permalink:
http://dlmf.nist.gov/33.4.E4
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§33.4
and
Ch.33