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NIST
14
Legendre and Related Functions
Real Arguments
14.3
Definitions and Hypergeometric Representations
14.5
Special Values
§14.4
Graphics
ⓘ
Permalink:
http://dlmf.nist.gov/14.4
See also:
Annotations for
Ch.14
Contents
§14.4(i)
Ferrers Functions: 2D Graphs
§14.4(ii)
Ferrers Functions: 3D Surfaces
§14.4(iii)
Associated Legendre Functions: 2D Graphs
§14.4(iv)
Associated Legendre Functions: 3D Surfaces
§14.4(i)
Ferrers Functions: 2D Graphs
ⓘ
Keywords:
Ferrers functions
,
graphics
Notes:
These graphs were produced at NIST.
Permalink:
http://dlmf.nist.gov/14.4.i
See also:
Annotations for
§14.4
and
Ch.14
Figure 14.4.1:
P
ν
0
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
x
)
: Ferrers function of the first kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F1
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
Figure 14.4.2:
Q
ν
0
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
x
)
: Ferrers function of the second kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F2
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
Figure 14.4.3:
P
ν
-
1
/
2
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
x
)
: Ferrers function of the first kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F3
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
Figure 14.4.4:
Q
ν
1
/
2
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
x
)
: Ferrers function of the second kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F4
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
Figure 14.4.5:
P
ν
-
1
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
x
)
: Ferrers function of the first kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F5
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
Figure 14.4.6:
Q
ν
1
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
x
)
: Ferrers function of the second kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F6
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
Figure 14.4.7:
P
0
-
μ
(
x
)
,
μ
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
x
)
: Ferrers function of the first kind
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F7
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
Figure 14.4.8:
Q
0
μ
(
x
)
,
μ
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
x
)
: Ferrers function of the second kind
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F8
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
Figure 14.4.9:
P
1
/
2
-
μ
(
x
)
,
μ
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
x
)
: Ferrers function of the first kind
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F9
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
Figure 14.4.10:
Q
1
/
2
μ
(
x
)
,
μ
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
x
)
: Ferrers function of the second kind
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F10
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
Figure 14.4.11:
P
1
-
μ
(
x
)
,
μ
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
x
)
: Ferrers function of the first kind
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F11
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
Figure 14.4.12:
Q
1
μ
(
x
)
,
μ
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
x
)
: Ferrers function of the second kind
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F12
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(i)
,
§14.4
and
Ch.14
§14.4(ii)
Ferrers Functions: 3D Surfaces
ⓘ
Keywords:
Ferrers functions
,
graphics
Notes:
These surfaces were produced at NIST.
Permalink:
http://dlmf.nist.gov/14.4.ii
See also:
Annotations for
§14.4
and
Ch.14
Figure 14.4.13:
P
ν
0
(
x
)
,
0
≤
ν
≤
10
,
-
1
<
x
<
1
.
Magnify
3D
Help
ⓘ
Symbols:
P
ν
μ
(
x
)
: Ferrers function of the first kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F13
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§14.4(ii)
,
§14.4
and
Ch.14
Figure 14.4.14:
Q
ν
0
(
x
)
,
0
≤
ν
≤
10
,
-
1
<
x
<
1
.
Magnify
3D
Help
ⓘ
Symbols:
Q
ν
μ
(
x
)
: Ferrers function of the second kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F14
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§14.4(ii)
,
§14.4
and
Ch.14
Figure 14.4.15:
P
0
-
μ
(
x
)
,
0
≤
μ
≤
10
,
-
1
<
x
<
1
.
Magnify
3D
Help
ⓘ
Symbols:
P
ν
μ
(
x
)
: Ferrers function of the first kind
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F15
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§14.4(ii)
,
§14.4
and
Ch.14
Figure 14.4.16:
Q
0
μ
(
x
)
,
0
≤
μ
≤
6.2
,
-
1
<
x
<
1
.
Magnify
3D
Help
ⓘ
Symbols:
Q
ν
μ
(
x
)
: Ferrers function of the second kind
,
x
: real variable
and
μ
: general order
Keywords:
Ferrers functions
,
graphics
Permalink:
http://dlmf.nist.gov/14.4.F16
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§14.4(ii)
,
§14.4
and
Ch.14
§14.4(iii)
Associated Legendre Functions: 2D Graphs
ⓘ
Keywords:
associated Legendre functions
,
graphics
Notes:
These graphs were produced at NIST.
Permalink:
http://dlmf.nist.gov/14.4.iii
See also:
Annotations for
§14.4
and
Ch.14
Figure 14.4.17:
P
ν
0
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
z
)
: associated Legendre function of the first kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F17
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
Figure 14.4.18:
Q
ν
0
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
z
)
: Olver’s associated Legendre function
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F18
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
Figure 14.4.19:
P
ν
-
1
/
2
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
z
)
: associated Legendre function of the first kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F19
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
Figure 14.4.20:
Q
ν
1
/
2
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
z
)
: Olver’s associated Legendre function
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F20
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
Figure 14.4.21:
P
ν
-
1
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
z
)
: associated Legendre function of the first kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F21
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
Figure 14.4.22:
Q
ν
1
(
x
)
,
ν
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
z
)
: Olver’s associated Legendre function
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F22
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
Figure 14.4.23:
P
0
-
μ
(
x
)
,
μ
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
z
)
: associated Legendre function of the first kind
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F23
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
Figure 14.4.24:
Q
0
μ
(
x
)
,
μ
=
0
,
2
,
4
,
8
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
z
)
: Olver’s associated Legendre function
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F24
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
Figure 14.4.25:
P
1
/
2
-
μ
(
x
)
,
μ
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
z
)
: associated Legendre function of the first kind
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F25
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
Figure 14.4.26:
Q
1
/
2
μ
(
x
)
,
μ
=
0
,
2
,
4
,
8
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
z
)
: Olver’s associated Legendre function
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F26
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
Figure 14.4.27:
P
1
-
μ
(
x
)
,
μ
=
0
,
1
2
,
1
,
2
,
4
.
Magnify
ⓘ
Symbols:
P
ν
μ
(
z
)
: associated Legendre function of the first kind
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F27
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
Figure 14.4.28:
Q
1
μ
(
x
)
,
μ
=
0
,
2
,
4
,
8
.
Magnify
ⓘ
Symbols:
Q
ν
μ
(
z
)
: Olver’s associated Legendre function
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F28
Encodings:
pdf
,
png
See also:
Annotations for
§14.4(iii)
,
§14.4
and
Ch.14
§14.4(iv)
Associated Legendre Functions: 3D Surfaces
ⓘ
Keywords:
associated Legendre functions
,
graphics
Notes:
These surfaces were produced at NIST.
Permalink:
http://dlmf.nist.gov/14.4.iv
See also:
Annotations for
§14.4
and
Ch.14
Figure 14.4.29:
P
ν
0
(
x
)
,
0
≤
ν
≤
10
,
1
<
x
<
10
.
Magnify
3D
Help
ⓘ
Symbols:
P
ν
μ
(
z
)
: associated Legendre function of the first kind
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F29
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§14.4(iv)
,
§14.4
and
Ch.14
Figure 14.4.30:
Q
ν
0
(
x
)
,
0
≤
ν
≤
10
,
1
<
x
<
10
.
Magnify
3D
Help
ⓘ
Symbols:
Q
ν
μ
(
z
)
: Olver’s associated Legendre function
,
x
: real variable
and
ν
: general degree
Permalink:
http://dlmf.nist.gov/14.4.F30
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§14.4(iv)
,
§14.4
and
Ch.14
Figure 14.4.31:
P
0
-
μ
(
x
)
,
0
≤
μ
≤
10
,
1
<
x
<
10
.
Magnify
3D
Help
ⓘ
Symbols:
P
ν
μ
(
z
)
: associated Legendre function of the first kind
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F31
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§14.4(iv)
,
§14.4
and
Ch.14
Figure 14.4.32:
Q
0
μ
(
x
)
,
0
≤
μ
≤
10
,
1
<
x
<
10
.
Magnify
3D
Help
ⓘ
Symbols:
Q
ν
μ
(
z
)
: Olver’s associated Legendre function
,
x
: real variable
and
μ
: general order
Permalink:
http://dlmf.nist.gov/14.4.F32
Encodings:
Vizualization
,
pdf
,
png
See also:
Annotations for
§14.4(iv)
,
§14.4
and
Ch.14