When associated Legendre polynomials and associated Legendre , also associated spherical functions called, is functions , which in mathematics and theoretical physics are used. Since not all assigned Legendre polynomials are really polynomials , many authors also speak of assigned or associated Legendre functions .
The assigned Legendre polynomials are the solutions to the general Legendre equation:
(
1
-
x
2
)
d
2
y
d
x
2
-
2
x
d
y
d
x
+
(
ℓ
(
ℓ
+
1
)
-
m
2
1
-
x
2
)
y
=
0
{\ displaystyle (1-x ^ {2}) \, {\ frac {\ mathrm {d} ^ {2} \, y} {\ mathrm {d} x ^ {2}}} - 2x {\ frac { \ mathrm {d} y} {\ mathrm {d} x}} + \ left (\ ell (\ ell +1) - {\ frac {m ^ {2}} {1-x ^ {2}}} \ right) \, y = 0}
This ordinary differential equation has non-singular solutions in the interval only if and are integral with .
[
-
1
,
1
]
{\ displaystyle [-1.1]}
ℓ
{\ displaystyle \ ell \,}
m
{\ displaystyle m \,}
0
≤
m
≤
ℓ
{\ displaystyle 0 \ leq m \ leq \ ell}
The general Legendre equation (and thus the associated Legendre polynomials) is often encountered in physics, especially when there is spherical symmetry , such as in the central potential . Here the Laplace equation and related partial differential equations can often be reduced to the general Legendre equation . The most prominent example of this is the quantum mechanical solution of the energy states of the hydrogen atom .
definition
The assigned Legendre polynomials for
m = 0 are the usual Legendre polynomials.
Assigned Legendre polynomials for
m = 1
Assigned Legendre polynomials for
m = 2
Assigned Legendre polynomials for
m = 3
The associated Legendre polynomials are referred to as . The easiest way to define them as derivatives of ordinary Legendre polynomials :
P
ℓ
(
m
)
(
x
)
{\ displaystyle P _ {\ ell} ^ {(m)} (x)}
P
ℓ
(
m
)
(
x
)
=
(
-
1
)
m
(
1
-
x
2
)
m
/
2
d
m
d
x
m
P
ℓ
(
x
)
{\ displaystyle P _ {\ ell} ^ {(m)} (x) = (- 1) ^ {m} \ left (1-x ^ {2} \ right) ^ {m / 2} {\ frac {\ mathrm {d} ^ {m}} {\ mathrm {d} x ^ {m}}} P _ {\ ell} (x)}
where the -th is Legendre polynomial
P
ℓ
(
x
)
{\ displaystyle P _ {\ ell} (x)}
ℓ
{\ displaystyle \ ell}
P
ℓ
(
x
)
=
1
2
ℓ
ℓ
!
d
ℓ
d
x
ℓ
(
x
2
-
1
)
ℓ
{\ displaystyle P _ {\ ell} (x) = {\ frac {1} {2 ^ {\ ell} \, \ ell!}} \, {\ frac {\ mathrm {d} ^ {\ ell}} { \ mathrm {d} x ^ {\ ell}}} \ left (x ^ {2} -1 \ right) ^ {\ ell}}
.
This results in
P
ℓ
(
m
)
(
x
)
=
(
-
1
)
m
2
ℓ
ℓ
!
(
1
-
x
2
)
m
/
2
d
ℓ
+
m
d
x
ℓ
+
m
(
x
2
-
1
)
ℓ
.
{\ displaystyle P _ {\ ell} ^ {(m)} (x) = {\ frac {(-1) ^ {m}} {2 ^ {\ ell} \, \ ell!}} \ left (1- x ^ {2} \ right) ^ {m / 2} {\ frac {\ mathrm {d} ^ {\ ell + m}} {\ mathrm {d} x ^ {\ ell + m}}} \ left ( x ^ {2} -1 \ right) ^ {\ ell}.}
Connection with Legendre polynomials
The generalized Legendre equation goes over into the Legendre equation so that .
m
=
0
{\ displaystyle m = 0}
P
ℓ
(
0
)
(
x
)
=
P
ℓ
(
x
)
{\ displaystyle P _ {\ ell} ^ {(0)} (x) = P _ {\ ell} (x)}
Orthogonality
For the assigned Legendre polynomials, two orthogonality relations apply in the interval :
I.
=
[
-
1
,
1
]
{\ displaystyle I = [- 1,1]}
∫
-
1
+
1
P
ℓ
(
m
)
(
x
)
P
k
(
m
)
(
x
)
d
x
=
2
2
ℓ
+
1
(
ℓ
+
m
)
!
(
ℓ
-
m
)
!
δ
ℓ
k
.
{\ displaystyle \ int \ limits _ {- 1} ^ {+ 1} P _ {\ ell} ^ {(m)} (x) \, P_ {k} ^ {(m)} (x) \, \ mathrm {d} x = {\ frac {2} {2 \, \ ell +1}} \, {\ frac {(\ ell + m)!} {(\ ell -m)!}} \, \ delta _ {\ ell k}.}
∫
-
1
+
1
P
ℓ
(
m
)
(
x
)
P
ℓ
(
n
)
(
x
)
⋅
1
1
-
x
2
d
x
=
(
ℓ
+
m
)
!
m
(
ℓ
-
m
)
!
δ
m
n
.
{\ displaystyle \ int \ limits _ {- 1} ^ {+ 1} P _ {\ ell} ^ {(m)} (x) \, P _ {\ ell} ^ {(n)} (x) \ cdot { \ frac {1} {1-x ^ {2}}} \, \ mathrm {d} x = {\ frac {(\ ell + m)!} {m (\ ell -m)!}} \, \ delta _ {mn}.}
The second integral is only defined if either or not equal to 0.
m
{\ displaystyle m}
n
{\ displaystyle n}
Connection with the unit sphere
Most important is the case . The associated Legendre equation then reads
x
=
cos
ϑ
{\ displaystyle x = \ cos \ vartheta}
d
2
y
d
ϑ
2
+
cos
ϑ
sin
ϑ
d
y
d
ϑ
+
[
ℓ
(
ℓ
+
1
)
-
m
2
sin
2
ϑ
]
y
=
0.
{\ displaystyle {\ frac {\ mathrm {d} ^ {2} y} {\ mathrm {d} \ vartheta ^ {2}}} + {\ frac {\ cos \ vartheta} {\ sin \ vartheta}} \ , {\ frac {\ mathrm {d} y} {\ mathrm {d} \ vartheta}} + \ left [\ ell \, (\ ell +1) - {\ frac {m ^ {2}} {\ sin ^ {2} \ vartheta}} \ right] y = 0.}
Because according to the substitution rule
∫
0
π
f
(
cos
ϑ
)
sin
ϑ
d
ϑ
=
∫
-
1
1
f
(
x
)
d
x
{\ displaystyle \ int _ {0} ^ {\ pi} f (\ cos \ vartheta) \ sin \ vartheta \, \ mathrm {d} \ vartheta = \ int _ {- 1} ^ {1} f (x) \ mathrm {d} x}
holds, the above orthogonality relations are easily transferred to the unit sphere.
The so-called spherical surface functions are defined as
P
ℓ
(
m
)
(
cos
ϑ
)
{\ displaystyle P _ {\ ell} ^ {(m)} (\ cos \ vartheta)}
Y
ℓ
(
m
)
(
φ
,
ϑ
)
=
2
ℓ
+
1
4th
π
(
ℓ
-
m
)
!
(
ℓ
+
m
)
!
P
ℓ
(
m
)
(
cos
ϑ
)
e
i
m
φ
,
{\ displaystyle Y _ {\ ell} ^ {(m)} (\ varphi, \ vartheta) = {\ sqrt {{\ frac {2 \, \ ell +1} {4 \, \ pi}} \, {\ frac {(\ ell -m)!} {(\ ell + m)!}}}} \, P _ {\ ell} ^ {(m)} (\ cos \ vartheta) \, \ mathrm {e} ^ { i \, m \, \ varphi},}
which form a complete orthonormal system on the unit sphere.
The first assigned Legendre polynomials
The following recursion formula applies to the assigned Legendre polynomials
(
ℓ
-
m
)
P
ℓ
(
m
)
(
x
)
=
x
(
2
ℓ
-
1
)
P
ℓ
-
1
(
m
)
(
x
)
-
(
ℓ
+
m
-
1
)
P
ℓ
-
2
(
m
)
(
x
)
.
{\ displaystyle (\ ell -m) \, P _ {\ ell} ^ {(m)} (x) = x \, (2 \, \ ell -1) \, P _ {\ ell -1} ^ {( m)} (x) - (\ ell + m-1) \, P _ {\ ell -2} ^ {(m)} (x).}
The corresponding start values of the recursion formula are shown as follows:
P
m
(
m
)
(
x
)
=
(
-
1
)
m
⋅
(
2
m
)
!
2
m
m
!
⋅
(
1
-
x
2
)
m
/
2
,
P
k
m
(
x
)
=
0
,
∀
k
<
m
{\ displaystyle P_ {m} ^ {(m)} (x) = (- 1) ^ {m} \ cdot {\ frac {(2m)!} {2 ^ {m} m!}} \ cdot \ left (1-x ^ {2} \ right) ^ {m / 2} \ quad, \ quad P_ {k} ^ {m} (x) = 0 \;, \ quad \ forall k <m}
The first Legendre polynomials are thus determined
P
0
(
0
)
(
x
)
=
1
{\ displaystyle P_ {0} ^ {(0)} (x) = 1 \! \,}
P
1
(
0
)
(
x
)
=
x
{\ displaystyle P_ {1} ^ {(0)} (x) = x \! \,}
P
1
(
1
)
(
x
)
=
-
1
-
x
2
{\ displaystyle P_ {1} ^ {(1)} (x) = - {\ sqrt {1-x ^ {2}}}}
P
2
(
0
)
(
x
)
=
1
2
(
3
x
2
-
1
)
{\ displaystyle P_ {2} ^ {(0)} (x) = {\ frac {1} {2}} \, (3 \, x ^ {2} -1)}
P
2
(
1
)
(
x
)
=
-
3
x
1
-
x
2
{\ displaystyle P_ {2} ^ {(1)} (x) = - 3 \, x {\ sqrt {1-x ^ {2}}}}
P
2
(
2
)
(
x
)
=
3
(
1
-
x
2
)
{\ displaystyle P_ {2} ^ {(2)} (x) = 3 \, (1-x ^ {2})}
And with as an argument
cos
ϑ
{\ displaystyle \ cos \ vartheta}
P
0
(
0
)
(
cos
ϑ
)
=
1
{\ displaystyle P_ {0} ^ {(0)} (\ cos \ vartheta) = 1}
P
1
(
0
)
(
cos
ϑ
)
=
cos
ϑ
{\ displaystyle P_ {1} ^ {(0)} (\ cos \ vartheta) = \ cos \ vartheta}
P
1
(
1
)
(
cos
ϑ
)
=
-
sin
ϑ
{\ displaystyle P_ {1} ^ {(1)} (\ cos \ vartheta) = - \ sin \ vartheta}
P
2
(
0
)
(
cos
ϑ
)
=
1
2
(
3
cos
2
ϑ
-
1
)
{\ displaystyle P_ {2} ^ {(0)} (\ cos \ vartheta) = {\ frac {1} {2}} \, (3 \, \ cos ^ {2} \ vartheta -1)}
P
2
(
1
)
(
cos
ϑ
)
=
-
3
sin
ϑ
cos
ϑ
{\ displaystyle P_ {2} ^ {(1)} (\ cos \ vartheta) = - 3 \, \ sin \ vartheta \, \ cos \ vartheta}
P
2
(
2
)
(
cos
ϑ
)
=
3
sin
2
ϑ
{\ displaystyle P_ {2} ^ {(2)} (\ cos \ vartheta) = 3 \, \ sin ^ {2} \ vartheta}
Assigned Legendre functions of the 2nd kind
Similar to the Legendre equation, the assigned Legendre polynomials represent only a group of solution functions of the generalized Legendre equation. The assigned Legendre functions of the 2nd type also represent solutions. The same applies to them with the Legendre functions of the 2nd type .
P
ℓ
(
m
)
(
x
)
{\ displaystyle P _ {\ ell} ^ {(m)} (x)}
Q
ℓ
(
m
)
(
x
)
{\ displaystyle Q _ {\ ell} ^ {(m)} (x)}
Q
ℓ
(
0
)
=
Q
ℓ
{\ displaystyle Q _ {\ ell} ^ {(0)} = Q _ {\ ell}}
Q
ℓ
(
x
)
{\ displaystyle Q _ {\ ell} (x)}
Web links
literature
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