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33
Coulomb Functions
Variables
Ο
,
Ξ·
33.2
Definitions and Basic Properties
33.4
Recurrence Relations and Derivatives
Β§33.3
Graphics
β
Keywords:
Coulomb functions: variables
Ο
,
Ξ·
,
graphics
Permalink:
http://dlmf.nist.gov/33.3
See also:
Annotations for
Ch.33
Contents
Β§33.3(i)
Line Graphs of the Coulomb Radial Functions
F
β
β‘
(
Ξ·
,
Ο
)
and
G
β
β‘
(
Ξ·
,
Ο
)
Β§33.3(ii)
Surfaces of the Coulomb Radial Functions
F
0
β‘
(
Ξ·
,
Ο
)
and
G
0
β‘
(
Ξ·
,
Ο
)
Β§33.3(i)
Line Graphs of the Coulomb Radial Functions
F
β
β‘
(
Ξ·
,
Ο
)
and
G
β
β‘
(
Ξ·
,
Ο
)
β
Notes:
These graphs were produced at NIST.
Permalink:
http://dlmf.nist.gov/33.3.i
See also:
Annotations for
Β§33.3
and
Ch.33
Figure 33.3.1:
F
β
β‘
(
Ξ·
,
Ο
)
,
G
β
β‘
(
Ξ·
,
Ο
)
with
β
=
0
,
Ξ·
=
β
2
.
Magnify
β
Symbols:
G
β
β‘
(
Ξ·
,
Ο
)
: irregular Coulomb radial function
,
F
β
β‘
(
Ξ·
,
Ο
)
: regular Coulomb radial function
,
β
: nonnegative integer
,
Ο
: nonnegative real variable
and
Ξ·
: real parameter
Permalink:
http://dlmf.nist.gov/33.3.F1
Encodings:
pdf
,
png
See also:
Annotations for
Β§33.3(i)
,
Β§33.3
and
Ch.33
Figure 33.3.2:
F
β
β‘
(
Ξ·
,
Ο
)
,
G
β
β‘
(
Ξ·
,
Ο
)
with
β
=
0
,
Ξ·
=
0
.
Magnify
β
Symbols:
G
β
β‘
(
Ξ·
,
Ο
)
: irregular Coulomb radial function
,
F
β
β‘
(
Ξ·
,
Ο
)
: regular Coulomb radial function
,
β
: nonnegative integer
,
Ο
: nonnegative real variable
and
Ξ·
: real parameter
Permalink:
http://dlmf.nist.gov/33.3.F2
Encodings:
pdf
,
png
See also:
Annotations for
Β§33.3(i)
,
Β§33.3
and
Ch.33
Figure 33.3.3:
F
β
β‘
(
Ξ·
,
Ο
)
,
G
β
β‘
(
Ξ·
,
Ο
)
with
β
=
0
,
Ξ·
=
2
. The turning point is at
Ο
tp
β‘
(
2
,
0
)
=
4
.
Magnify
β
Symbols:
Ο
tp
β‘
(
Ξ·
,
β
)
: outer turning point for Coulomb radial functions
,
G
β
β‘
(
Ξ·
,
Ο
)
: irregular Coulomb radial function
,
F
β
β‘
(
Ξ·
,
Ο
)
: regular Coulomb radial function
,
β
: nonnegative integer
,
Ο
: nonnegative real variable
and
Ξ·
: real parameter
Permalink:
http://dlmf.nist.gov/33.3.F3
Encodings:
pdf
,
png
See also:
Annotations for
Β§33.3(i)
,
Β§33.3
and
Ch.33
Figure 33.3.4:
F
β
β‘
(
Ξ·
,
Ο
)
,
G
β
β‘
(
Ξ·
,
Ο
)
with
β
=
0
,
Ξ·
=
10
. The turning point is at
Ο
tp
β‘
(
10
,
0
)
=
20
.
Magnify
β
Symbols:
Ο
tp
β‘
(
Ξ·
,
β
)
: outer turning point for Coulomb radial functions
,
G
β
β‘
(
Ξ·
,
Ο
)
: irregular Coulomb radial function
,
F
β
β‘
(
Ξ·
,
Ο
)
: regular Coulomb radial function
,
β
: nonnegative integer
,
Ο
: nonnegative real variable
and
Ξ·
: real parameter
Permalink:
http://dlmf.nist.gov/33.3.F4
Encodings:
pdf
,
png
See also:
Annotations for
Β§33.3(i)
,
Β§33.3
and
Ch.33
In Figures
33.3.5
and
33.3.6
33.3.1
M
β
β‘
(
Ξ·
,
Ο
)
=
(
F
β
2
β‘
(
Ξ·
,
Ο
)
+
G
β
2
β‘
(
Ξ·
,
Ο
)
)
1
/
2
=
|
H
β
Β±
β‘
(
Ξ·
,
Ο
)
|
.
β
Defines:
M
β
β‘
(
Ξ·
,
Ο
)
: envelope of Coulomb functions
Symbols:
G
β
β‘
(
Ξ·
,
Ο
)
: irregular Coulomb radial function
,
H
β
Β±
β‘
(
Ξ·
,
Ο
)
: irregular Coulomb radial functions
,
F
β
β‘
(
Ξ·
,
Ο
)
: regular Coulomb radial function
,
β
: nonnegative integer
,
Ο
: nonnegative real variable
and
Ξ·
: real parameter
Permalink:
http://dlmf.nist.gov/33.3.E1
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
Β§33.3(i)
,
Β§33.3
and
Ch.33
Figure 33.3.5:
F
β
β‘
(
Ξ·
,
Ο
)
,
G
β
β‘
(
Ξ·
,
Ο
)
, and
M
β
β‘
(
Ξ·
,
Ο
)
with
β
=
0
,
Ξ·
=
15
/
2
. The turning point is at
Ο
tp
β‘
(
15
/
2
,
0
)
=
30
=
5.47
β’
β¦
.
Magnify
β
Symbols:
Ο
tp
β‘
(
Ξ·
,
β
)
: outer turning point for Coulomb radial functions
,
M
β
β‘
(
Ξ·
,
Ο
)
: envelope of Coulomb functions
,
G
β
β‘
(
Ξ·
,
Ο
)
: irregular Coulomb radial function
,
F
β
β‘
(
Ξ·
,
Ο
)
: regular Coulomb radial function
,
β
: nonnegative integer
,
Ο
: nonnegative real variable
and
Ξ·
: real parameter
Referenced by:
Figure 33.3.6
,
Figure 33.3.6
,
Β§33.3(i)
Permalink:
http://dlmf.nist.gov/33.3.F5
Encodings:
pdf
,
png
See also:
Annotations for
Β§33.3(i)
,
Β§33.3
and
Ch.33
Figure 33.3.6:
F
β
β‘
(
Ξ·
,
Ο
)
,
G
β
β‘
(
Ξ·
,
Ο
)
, and
M
β
β‘
(
Ξ·
,
Ο
)
with
β
=
5
,
Ξ·
=
0
. The turning point is at
Ο
tp
β‘
(
0
,
5
)
=
30
(as in Figure
33.3.5
).
Magnify
β
Symbols:
Ο
tp
β‘
(
Ξ·
,
β
)
: outer turning point for Coulomb radial functions
,
M
β
β‘
(
Ξ·
,
Ο
)
: envelope of Coulomb functions
,
G
β
β‘
(
Ξ·
,
Ο
)
: irregular Coulomb radial function
,
F
β
β‘
(
Ξ·
,
Ο
)
: regular Coulomb radial function
,
β
: nonnegative integer
,
Ο
: nonnegative real variable
and
Ξ·
: real parameter
Referenced by:
Β§33.3(i)
Permalink:
http://dlmf.nist.gov/33.3.F6
Encodings:
pdf
,
png
See also:
Annotations for
Β§33.3(i)
,
Β§33.3
and
Ch.33
Β§33.3(ii)
Surfaces of the Coulomb Radial Functions
F
0
β‘
(
Ξ·
,
Ο
)
and
G
0
β‘
(
Ξ·
,
Ο
)
β
Keywords:
Coulomb functions: variables
Ο
,
Ξ·
,
graphics
Notes:
These surfaces were produced at NIST.
Permalink:
http://dlmf.nist.gov/33.3.ii
See also:
Annotations for
Β§33.3
and
Ch.33
Figure 33.3.7:
F
0
β‘
(
Ξ·
,
Ο
)
,
β
2
β€
Ξ·
β€
2
,
0
β€
Ο
β€
5
.
Magnify
3D
Help
β
Symbols:
F
β
β‘
(
Ξ·
,
Ο
)
: regular Coulomb radial function
,
Ο
: nonnegative real variable
and
Ξ·
: real parameter
Permalink:
http://dlmf.nist.gov/33.3.F7
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
Β§33.3(ii)
,
Β§33.3
and
Ch.33
Figure 33.3.8:
G
0
β‘
(
Ξ·
,
Ο
)
,
β
2
β€
Ξ·
β€
2
,
0
<
Ο
β€
5
.
Magnify
3D
Help
β
Symbols:
G
β
β‘
(
Ξ·
,
Ο
)
: irregular Coulomb radial function
,
Ο
: nonnegative real variable
and
Ξ·
: real parameter
Permalink:
http://dlmf.nist.gov/33.3.F8
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
Β§33.3(ii)
,
Β§33.3
and
Ch.33