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NIST
33
Coulomb Functions
Variables
r
,
ϵ
33.16
Connection Formulas
33.18
Limiting Forms for Large
ℓ
§33.17
Recurrence Relations and Derivatives
ⓘ
Keywords:
Coulomb functions: variables
r
,
ϵ
,
derivatives
,
recurrence relations
Notes:
See
Seaton (
2002a
, Eqs. 77, 78, 82)
.
Referenced by:
§33.23(iv)
,
§33.23(iv)
Permalink:
http://dlmf.nist.gov/33.17
See also:
Annotations for
Ch.33
33.17.1
(
ℓ
+
1
)
r
f
(
ϵ
,
ℓ
-
1
;
r
)
-
(
2
ℓ
+
1
)
(
ℓ
(
ℓ
+
1
)
-
r
)
f
(
ϵ
,
ℓ
;
r
)
+
ℓ
(
1
+
(
ℓ
+
1
)
2
ϵ
)
r
f
(
ϵ
,
ℓ
+
1
;
r
)
=
0
,
ⓘ
Symbols:
f
(
ϵ
,
ℓ
;
r
)
: regular Coulomb function
,
ℓ
: nonnegative integer
,
r
: real variable
and
ϵ
: real parameter
Permalink:
http://dlmf.nist.gov/33.17.E1
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§33.17
and
Ch.33
33.17.2
(
ℓ
+
1
)
(
1
+
ℓ
2
ϵ
)
r
h
(
ϵ
,
ℓ
-
1
;
r
)
-
(
2
ℓ
+
1
)
(
ℓ
(
ℓ
+
1
)
-
r
)
h
(
ϵ
,
ℓ
;
r
)
+
ℓ
r
h
(
ϵ
,
ℓ
+
1
;
r
)
=
0
,
ⓘ
Symbols:
h
(
ϵ
,
ℓ
;
r
)
: irregular Coulomb function
,
ℓ
: nonnegative integer
,
r
: real variable
and
ϵ
: real parameter
Permalink:
http://dlmf.nist.gov/33.17.E2
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§33.17
and
Ch.33
33.17.3
(
ℓ
+
1
)
r
f
′
(
ϵ
,
ℓ
;
r
)
=
(
(
ℓ
+
1
)
2
-
r
)
f
(
ϵ
,
ℓ
;
r
)
-
(
1
+
(
ℓ
+
1
)
2
ϵ
)
r
f
(
ϵ
,
ℓ
+
1
;
r
)
,
ⓘ
Symbols:
f
(
ϵ
,
ℓ
;
r
)
: regular Coulomb function
,
ℓ
: nonnegative integer
,
r
: real variable
and
ϵ
: real parameter
Permalink:
http://dlmf.nist.gov/33.17.E3
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§33.17
and
Ch.33
33.17.4
(
ℓ
+
1
)
r
h
′
(
ϵ
,
ℓ
;
r
)
=
(
(
ℓ
+
1
)
2
-
r
)
h
(
ϵ
,
ℓ
;
r
)
-
r
h
(
ϵ
,
ℓ
+
1
;
r
)
.
ⓘ
Symbols:
h
(
ϵ
,
ℓ
;
r
)
: irregular Coulomb function
,
ℓ
: nonnegative integer
,
r
: real variable
and
ϵ
: real parameter
Permalink:
http://dlmf.nist.gov/33.17.E4
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§33.17
and
Ch.33