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SOME PROPERTIES OF THE LEGENDRE POLYNOMIALS
A P P E N D I X III
S O M E PROPERTIES O F THE L E G E N D R E P O L Y N O M I A L S
§1. The generating function and recursion formulas The Legendre polynomials P„(cos Θ) were defined in section 3 by the equation: F(x, z) =
= f
1
yj\
— 2XZ
+
P (x)z",
(A3.1)
n
n= 0
Ζ
where χ = cos θ and 0 < ζ < 1. The function F(x, z) is called the generating function. With the help of the generating function three well-known recursion formulas for the Legendre polynomials can be derived. First we differentiate (A3.1) with respect to ζ:
l = - ^ r f ? F = i" " ""' p
·
Wz
(Α3 2)
Substituting (A3.1), we o b t a i n : - ( z - x) f
P„(x)z" = (1 - 2xz + ζ ) £
nP^xV-K
2
n=0
(A3.3)
n=0
F o r each value of n, the coefficients of z on b o t h sides must be equal. Thus, we find for the coefficient of z : n
n
- Ρ , - Λ χ ) + xP (x) = (n+ n
l ) P „ ( x ) - 2nxP (x) +1
n
+ (>i -
l)P - (x), n x
or: (n + l ) P
M+1
( x ) - {In + l)xP (x) n
+ nP .,(x) n
= 0.
(A3.4)
With the help of this recursion formula all Legendre polynomials can be expressed in terms of the first two, e.g.: Ρ (χ) = *[3*Ρι(χ) 2
P (x)l 0
P W = lC5x{|xPi(x) - i P W } 3
0
= Ml5x
2
2Ρ (χ)] χ
- 4)Λ(χ) - fxPo(x)].
358
C . J. F . BOTTCHER
W e can also differentiate the generating function with respect to x, obtaining: - dx (1-2χζ
~ V ^ z " % dx
Z
+ ζψ
2
K
(A35) }
n
Substitution of (A3.1) now leads t o : dP(x) ζ Σ PJixW = (1 - 2xz + ζ ) Σ -f ^. 00
00
2
Comparing coefficients of z
dP„(x)
2x—f~±
+1
x
"
2
, we find:
dP„ (x) ' dx
p ( ) = n
n + 1
(A3.6)
1
n=0
n=0
dx
dP„ ,(x) + _szlLZ. dx
(A3.7)
We can also compare (A3.2) and (A3.5), and e q u a t e : * dPix) = (χ - ζ) Σ ~Τ~^ "·
00
ζ Σ ηΡ {χ)ζ"-
1
(Α3.8)
Ζ
Λ
π=0
η=0
α
χ
Equating coefficients of ζ we o b t a i n : η
<
(
X
)
=
^ M _ M .
X
dx
dx
( A
,
( A 3
.
9 )
d P (x) Eliminating x — ^ — from (A3.7) and (A3.9), we find: dx ( 2
„ + 1)F„
W
= % i W dx
-
1 0 )
dx
§2. Legendre's differential equation The Legendre polynomials, defined by (A3.1), satisfy a differential equation called Legendre's differential equation. This can be shown with the help of the recursion formulas (A3.9) and (A3.10). Differentiating (A3.9) with respect to x, we find: n
dP„(x) dx
ά Ρ (χ) dx 2
=
χ
η
+ 2
dP^x) _ d ' P . - . f r ) dx dx 2
Raising in eqn. (A3.11) the index η to (n + 1) we also h a v e :
(
A
3
n
)
359
SOME PROPERTIES OF THE LEGENDRE POLYNOMIALS
dP„ (x) d P„ (x) dP (x) d P„(x) (n + 1 " = χ — + " - 4 - ^ · (A3.12) dx dx dx ax If we now multiply (A3.ll) through with x, and add the result to (A3.12), we h a v e : 2
2
+ 1 1
+
1
n + 1
+ 1 V
2
dx
1
dx
T h e last term can be simplified, since we note that it is equal to χ times the derivative of the righthand side of (A3.10). Substituting this and rearranging, we o b t a i n :
a
_
^ _ dx"
x
(
„
+
2
)
^W dx
x
+
„%iW dx
0.
=
(A3.14)
dP + (x) dP (x) — ^ — can be expressed in terms of P (x) and — j — by combining (A3.10) dx ax and (A3.9), which leads t o : n
dP„+i(x) , . , ν , dP (x) — = (n+ l)P (x) + x— dx dx Substituting this into (A3.14), we finally arrive a t : n
D
« (A3.15)
rt
/
n
(1 - x )^Pdx
- 2χ ζ^dx
2
ά
+ n(n + l)P (x) = 0. n
A
1
1
(A3.16)
This is Legendre's differential e q u a t i o n : (1 - x
2
) ^ - 2x^
+ n(n + l)y = 0,
or: _d_ dx
(1 - x ) - ^ 2
+ n(n + l)y = 0.
(A3.17)
F r o m (A3.16) it follows that the Legendre polynomials P are particular solutions of the Legendre equation when η is an integer. n
360
C. J. F. BOTTCHER
§3. The solution of Legendre's equation The general solution of Legendre's equation on the interval — 1 ^ x ^ + 1 can be found by assuming the solution y(x) to be expressed as a power series in x : y(x)
= Σ
(A3.18)
j-
c xi
j=o
Provided that this series converges, we can substitute it into Legendre's equation (A3.17) to find a relation for the coefficients Cj. W e shall write ν instead of η in (A3.17) so as not to prejudge the issue of whether or not ν is an integer. When the differentiations have been performed on the power series, we o b t a i n :
00
CO
Σ
JU ~ l ) ^ ' "
2
-
Σ
{JU - 1) + 2/ - v(v +
= 0.
\)}cjx>
(A3.19)
7=0
j=0
Equating the coefficients of the same power in χ we obtain the recurrence relation: j(j + 1) - v(v + 1) (A3.20) j +2 — 0' + DU + 2) ' C
Thus, all coefficients with even indices may be expressed in terms of c , and all coefficients with odd indices in terms of c : 0
x
\
y(x) = c
Q
, - v ( v + l) , 6-v(v+l) 1Η χ Η — 12 2
x +
2 - v(v + 1) -x
3 J
+
-v(v+l)
. χ
4
+
+
12 - v(v + 1) 2 - v(v + 1) 20
x
s
+
•(A3.21)
In this way the general solution of Legendre's equation is obtained as a linear combination of two power series in χ with two arbitrary constants c and C j . Each series converges when | x | < 1; when χ = ± 1 , however, both series will generally diverge, and the solution will have a singularity for these values of x . F r o m the recurrence relation (A3.20) it follows that c will become zero for some when j = v, i.e. when ν is an integer n. All higher coefficients c , etc. will then also be zero, so that the series terminates at the H-th term and there will be no question of divergence whatever value χ may 0
J+2
j+6
SOME PROPERTIES OF THE LEGENDRE
361
POLYNOMIALS
have. When η is even, the power series with coefficients c will reduce to a polynomial in χ and the power series with coefficient c will continue to diverge for χ = ± 1 . When η is odd, it will be the other way around. T h u s if a solution of Legendre's equation is required that has n o singu larity for χ = ± 1 , one has to take ν = η and either c or c equal to zero, depending on whether η is even or odd. The solutions obtained in this way are the Legendre polynomials defined in eqn. (A3.1) with the help of a generating function. F r o m this it immediately follows that P (x) is even or odd in χ when η is even or odd, respectively. When instead c or c are taken equal to zero for η odd or even, respectively, a power series in χ is obtained that diverges for χ = + 1 . This power series is called the Legendre function of the second k i n d and denoted by Q {x) (see also eqns. (A2.10) and (4.49)). 0
x
1
0
n
x
0
1
n
§4. A table of the Legendre polynomials With the help of (A3.20), taking c orc equal to 0 as required, all Legendre polynomials can be calculated explicitly. It is also possible to use the gener ating function defined in eqn. (A3.1), or, when the first two polynomials have been calculated, the recursion formula (A3.4). When a polynomial with a high value of η is required, it is easier to use the equation of Rodrigues: 0
p
"
- h
w
1
. s?
,
x
2
-
1
Γ
, Α 3
·
2 2 )
The result of the calculations for the first eight polynomials is the following table: Λ>(*) = 1 Pi(x) =
X
P (x) = ^(3x - 1) 2
2
P (x) = i ( 5 x - 3x) 3
3
Ρ*(χ) = i ( 3 5 x - 3 0 x + 3) 4
2
Psix) = i ( 6 3 x - 7 0 x + 15x) 5
3
Ρ (χ) = JV(231X - 3 1 5 x + 105x - 5) Pi(x) = xV429x - 693x + 315x - 35x) 6
6
7
4
2
5
3
362
C . J. F . BOTTCHER
§5. Orthogonality of the Legendre polynomials Two real functions Φ(χ) a n d Ψ(χ) are called orthogonal on an interval I when they satisfy the relation: j a > ( x M x ) d x = 0.
(A3.23)
ι The Legendre polynomials are orthogonal to each other on the interval — 1 ^ χ ^ + 1 . This can be shown with the help of the generating function Fix, z) (eqn. (A3.1)): 1
00
•Fix, ζ) =
=
ς
1 — 2xz + ζ
n=o
Using a different variable υ a n d a different index m, we may also write: 1
00
F(x, ν) =
= Σ ν 1 - 2xv + v
2
m
=
(A3.24) 0
Now J
J yj 1 - 2xz + z y l - 2xv + v
2
can be solved with the help of the relation:
^
log jy^l
+ z ) - Ivzx - yjz{\ + v ) 2
2
2vzx^
vz yjv{\
+ z ) - 2vzxjz{\ 2
+ v) 2
2vzx
Thus we find
f F(x, z)f (*,,) d x = J,
2
' lo, + ~ s/vz y/v(l + z ) + 2cz - ^ / z ( l + t> ) + 2cz 2
logi±X2-/(r
1
UZ
and
Μ + ° > ~ *»
^
+1
1
— /t;z x
2
A
(A3.25)
SOME PROPERTIES OF THE LEGENDRE
363
POLYNOMIALS
+1
/(«*)=Σ
Σ
m=0 n=0
I \ PJx)P (x)ax}v z".
(A3.26)
m
n
J
Since the lefthand side of (A3.26) is a function of the product vz only, develop ment in a power series will lead to terms of the form c v z . Therefore, the coefficients on the righthand side for m # η are zero, o r : +1 l
l
t
J P (x)P (x) άχ = 0 for m φ η. (A3.27) -1 Thus, the Legendre polynomials are orthogonal to each other on the interval - 1 < χ < +1. When m = n, we have, taking ν = ζ in (A3.25) and (A3.26): +1 m
n
Σ z
f {Ρ (χψ
2n
dx = - log
η
(A3.28)
-1 The series development of the righthand side with the help of the wellknown development for log (1 ± z) with |z| < 1 now leads t o : +1
£ z " f {P„U)} dx = 2
n=0
2
J
Σ
£ ζ Ί ·
n=0
(A3.29)
'
-1 Comparison of coefficients gives: +1 j * {P„(x)Pdx = ^ (A3.30) Eqns. (A3.27) and (A3.30) can now be expressed by one relation: -1 +1 r
T
J PJx)P (x)dx = 2^TT*«»' -1 where <5 , the Kronecker delta, is equal to zero ίοντηΦη for m = n.
(
A
3
M
)
n
mn
and equal to one
Reference 1. E. W. H o b s o n , The Theory of Spherical bridge 1955.
and Ellipsoidal
Harmonics,
University Press, C a m
S O M E PROPERTIES O F THE L E G E N D R E P O L Y N O M I A L S
§1. The generating function and recursion formulas The Legendre polynomials P„(cos Θ) were defined in section 3 by the equation: F(x, z) =
= f
1
yj\
— 2XZ
+
P (x)z",
(A3.1)
n
n= 0
Ζ
where χ = cos θ and 0 < ζ < 1. The function F(x, z) is called the generating function. With the help of the generating function three well-known recursion formulas for the Legendre polynomials can be derived. First we differentiate (A3.1) with respect to ζ:
l = - ^ r f ? F = i" " ""' p
·
Wz
(Α3 2)
Substituting (A3.1), we o b t a i n : - ( z - x) f
P„(x)z" = (1 - 2xz + ζ ) £
nP^xV-K
2
n=0
(A3.3)
n=0
F o r each value of n, the coefficients of z on b o t h sides must be equal. Thus, we find for the coefficient of z : n
n
- Ρ , - Λ χ ) + xP (x) = (n+ n
l ) P „ ( x ) - 2nxP (x) +1
n
+ (>i -
l)P - (x), n x
or: (n + l ) P
M+1
( x ) - {In + l)xP (x) n
+ nP .,(x) n
= 0.
(A3.4)
With the help of this recursion formula all Legendre polynomials can be expressed in terms of the first two, e.g.: Ρ (χ) = *[3*Ρι(χ) 2
P (x)l 0
P W = lC5x{|xPi(x) - i P W } 3
0
= Ml5x
2
2Ρ (χ)] χ
- 4)Λ(χ) - fxPo(x)].
358
C . J. F . BOTTCHER
W e can also differentiate the generating function with respect to x, obtaining: - dx (1-2χζ
~ V ^ z " % dx
Z
+ ζψ
2
K
(A35) }
n
Substitution of (A3.1) now leads t o : dP(x) ζ Σ PJixW = (1 - 2xz + ζ ) Σ -f ^. 00
00
2
Comparing coefficients of z
dP„(x)
2x—f~±
+1
x
"
2
, we find:
dP„ (x) ' dx
p ( ) = n
n + 1
(A3.6)
1
n=0
n=0
dx
dP„ ,(x) + _szlLZ. dx
(A3.7)
We can also compare (A3.2) and (A3.5), and e q u a t e : * dPix) = (χ - ζ) Σ ~Τ~^ "·
00
ζ Σ ηΡ {χ)ζ"-
1
(Α3.8)
Ζ
Λ
π=0
η=0
α
χ
Equating coefficients of ζ we o b t a i n : η
<
(
X
)
=
^ M _ M .
X
dx
dx
( A
,
( A 3
.
9 )
d P (x) Eliminating x — ^ — from (A3.7) and (A3.9), we find: dx ( 2
„ + 1)F„
W
= % i W dx
-
1 0 )
dx
§2. Legendre's differential equation The Legendre polynomials, defined by (A3.1), satisfy a differential equation called Legendre's differential equation. This can be shown with the help of the recursion formulas (A3.9) and (A3.10). Differentiating (A3.9) with respect to x, we find: n
dP„(x) dx
ά Ρ (χ) dx 2
=
χ
η
+ 2
dP^x) _ d ' P . - . f r ) dx dx 2
Raising in eqn. (A3.11) the index η to (n + 1) we also h a v e :
(
A
3
n
)
359
SOME PROPERTIES OF THE LEGENDRE POLYNOMIALS
dP„ (x) d P„ (x) dP (x) d P„(x) (n + 1 " = χ — + " - 4 - ^ · (A3.12) dx dx dx ax If we now multiply (A3.ll) through with x, and add the result to (A3.12), we h a v e : 2
2
+ 1 1
+
1
n + 1
+ 1 V
2
dx
1
dx
T h e last term can be simplified, since we note that it is equal to χ times the derivative of the righthand side of (A3.10). Substituting this and rearranging, we o b t a i n :
a
_
^ _ dx"
x
(
„
+
2
)
^W dx
x
+
„%iW dx
0.
=
(A3.14)
dP + (x) dP (x) — ^ — can be expressed in terms of P (x) and — j — by combining (A3.10) dx ax and (A3.9), which leads t o : n
dP„+i(x) , . , ν , dP (x) — = (n+ l)P (x) + x— dx dx Substituting this into (A3.14), we finally arrive a t : n
D
« (A3.15)
rt
/
n
(1 - x )^Pdx
- 2χ ζ^dx
2
ά
+ n(n + l)P (x) = 0. n
A
1
1
(A3.16)
This is Legendre's differential e q u a t i o n : (1 - x
2
) ^ - 2x^
+ n(n + l)y = 0,
or: _d_ dx
(1 - x ) - ^ 2
+ n(n + l)y = 0.
(A3.17)
F r o m (A3.16) it follows that the Legendre polynomials P are particular solutions of the Legendre equation when η is an integer. n
360
C. J. F. BOTTCHER
§3. The solution of Legendre's equation The general solution of Legendre's equation on the interval — 1 ^ x ^ + 1 can be found by assuming the solution y(x) to be expressed as a power series in x : y(x)
= Σ
(A3.18)
j-
c xi
j=o
Provided that this series converges, we can substitute it into Legendre's equation (A3.17) to find a relation for the coefficients Cj. W e shall write ν instead of η in (A3.17) so as not to prejudge the issue of whether or not ν is an integer. When the differentiations have been performed on the power series, we o b t a i n :
00
CO
Σ
JU ~ l ) ^ ' "
2
-
Σ
{JU - 1) + 2/ - v(v +
= 0.
\)}cjx>
(A3.19)
7=0
j=0
Equating the coefficients of the same power in χ we obtain the recurrence relation: j(j + 1) - v(v + 1) (A3.20) j +2 — 0' + DU + 2) ' C
Thus, all coefficients with even indices may be expressed in terms of c , and all coefficients with odd indices in terms of c : 0
x
\
y(x) = c
Q
, - v ( v + l) , 6-v(v+l) 1Η χ Η — 12 2
x +
2 - v(v + 1) -x
3 J
+
-v(v+l)
. χ
4
+
+
12 - v(v + 1) 2 - v(v + 1) 20
x
s
+
•(A3.21)
In this way the general solution of Legendre's equation is obtained as a linear combination of two power series in χ with two arbitrary constants c and C j . Each series converges when | x | < 1; when χ = ± 1 , however, both series will generally diverge, and the solution will have a singularity for these values of x . F r o m the recurrence relation (A3.20) it follows that c will become zero for some when j = v, i.e. when ν is an integer n. All higher coefficients c , etc. will then also be zero, so that the series terminates at the H-th term and there will be no question of divergence whatever value χ may 0
J+2
j+6
SOME PROPERTIES OF THE LEGENDRE
361
POLYNOMIALS
have. When η is even, the power series with coefficients c will reduce to a polynomial in χ and the power series with coefficient c will continue to diverge for χ = ± 1 . When η is odd, it will be the other way around. T h u s if a solution of Legendre's equation is required that has n o singu larity for χ = ± 1 , one has to take ν = η and either c or c equal to zero, depending on whether η is even or odd. The solutions obtained in this way are the Legendre polynomials defined in eqn. (A3.1) with the help of a generating function. F r o m this it immediately follows that P (x) is even or odd in χ when η is even or odd, respectively. When instead c or c are taken equal to zero for η odd or even, respectively, a power series in χ is obtained that diverges for χ = + 1 . This power series is called the Legendre function of the second k i n d and denoted by Q {x) (see also eqns. (A2.10) and (4.49)). 0
x
1
0
n
x
0
1
n
§4. A table of the Legendre polynomials With the help of (A3.20), taking c orc equal to 0 as required, all Legendre polynomials can be calculated explicitly. It is also possible to use the gener ating function defined in eqn. (A3.1), or, when the first two polynomials have been calculated, the recursion formula (A3.4). When a polynomial with a high value of η is required, it is easier to use the equation of Rodrigues: 0
p
"
- h
w
1
. s?
,
x
2
-
1
Γ
, Α 3
·
2 2 )
The result of the calculations for the first eight polynomials is the following table: Λ>(*) = 1 Pi(x) =
X
P (x) = ^(3x - 1) 2
2
P (x) = i ( 5 x - 3x) 3
3
Ρ*(χ) = i ( 3 5 x - 3 0 x + 3) 4
2
Psix) = i ( 6 3 x - 7 0 x + 15x) 5
3
Ρ (χ) = JV(231X - 3 1 5 x + 105x - 5) Pi(x) = xV429x - 693x + 315x - 35x) 6
6
7
4
2
5
3
362
C . J. F . BOTTCHER
§5. Orthogonality of the Legendre polynomials Two real functions Φ(χ) a n d Ψ(χ) are called orthogonal on an interval I when they satisfy the relation: j a > ( x M x ) d x = 0.
(A3.23)
ι The Legendre polynomials are orthogonal to each other on the interval — 1 ^ χ ^ + 1 . This can be shown with the help of the generating function Fix, z) (eqn. (A3.1)): 1
00
•Fix, ζ) =
=
ς
1 — 2xz + ζ
n=o
Using a different variable υ a n d a different index m, we may also write: 1
00
F(x, ν) =
= Σ ν 1 - 2xv + v
2
m
=
(A3.24) 0
Now J
J yj 1 - 2xz + z y l - 2xv + v
2
can be solved with the help of the relation:
^
log jy^l
+ z ) - Ivzx - yjz{\ + v ) 2
2
2vzx^
vz yjv{\
+ z ) - 2vzxjz{\ 2
+ v) 2
2vzx
Thus we find
f F(x, z)f (*,,) d x = J,
2
' lo, + ~ s/vz y/v(l + z ) + 2cz - ^ / z ( l + t> ) + 2cz 2
logi±X2-/(r
1
UZ
and
Μ + ° > ~ *»
^
+1
1
— /t;z x
2
A
(A3.25)
SOME PROPERTIES OF THE LEGENDRE
363
POLYNOMIALS
+1
/(«*)=Σ
Σ
m=0 n=0
I \ PJx)P (x)ax}v z".
(A3.26)
m
n
J
Since the lefthand side of (A3.26) is a function of the product vz only, develop ment in a power series will lead to terms of the form c v z . Therefore, the coefficients on the righthand side for m # η are zero, o r : +1 l
l
t
J P (x)P (x) άχ = 0 for m φ η. (A3.27) -1 Thus, the Legendre polynomials are orthogonal to each other on the interval - 1 < χ < +1. When m = n, we have, taking ν = ζ in (A3.25) and (A3.26): +1 m
n
Σ z
f {Ρ (χψ
2n
dx = - log
η
(A3.28)
-1 The series development of the righthand side with the help of the wellknown development for log (1 ± z) with |z| < 1 now leads t o : +1
£ z " f {P„U)} dx = 2
n=0
2
J
Σ
£ ζ Ί ·
n=0
(A3.29)
'
-1 Comparison of coefficients gives: +1 j * {P„(x)Pdx = ^ (A3.30) Eqns. (A3.27) and (A3.30) can now be expressed by one relation: -1 +1 r
T
J PJx)P (x)dx = 2^TT*«»' -1 where <5 , the Kronecker delta, is equal to zero ίοντηΦη for m = n.
(
A
3
M
)
n
mn
and equal to one
Reference 1. E. W. H o b s o n , The Theory of Spherical bridge 1955.
and Ellipsoidal
Harmonics,
University Press, C a m