10数理方法

16/54 Legendre
(10.1-17)
10.1.5
Legendre Fourier θ f ( x) =
Fourier
§ 9.4 Sturm-Livouville P l ( x) ( l = 0, 1, 2, · · · ) x [0, π]
§10.1. (10.1-13) 1 1+x Q0( x) = ln , 2 1−x 1 1+x − 1, Q1( x) = P1( x) ln 2 1−x 1 1+x 3 Q2( x) = P2( x) ln − x, 2 1−x 2 1+x 5 2 2 1 − x = , Q3( x) = p3( x) ln 2 1−x 2 3 1 1 + x 35 3 55 Q4( x) = P4( x) ln − x + x, 2 1−x 8 24 ············ Legendre y( x) = C1 P l ( x) + C2Q l ( x), C1 , C2 Q l ( x) x = ±1 Q l ( x)
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l( l + 1), l = 0, 1, 2, · · · m=0
(10.1-2)
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§10.1. a l−2 n = (−1) n l pl ( x) =
π 0
11/54 θ, x = cos θ
P l (−1) = (−1) l , | cos θ + i sin θ cos ψ| l dψ
(10.1-10)
π
cos2 θ + sin2 θ cos2 ψ
0 π
l/2
dψ
cos2 θ + sin2 θ
0 π
l /2
dψ
dψ = 1.
0
− 1 ≤ x ≤ 1.
(10.1-9)
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§10.1. Laplace x 1 π l P l ( x) = cos θ + i sin θ cos ψ dψ. π 0 (10.1-9) P l (1) = 1, 1 | P l ( x)| ≤ π 1 = π 1 ≤ π 1 = π | P l ( x)| ≤ 1,
k=0
·
n=0
(−1) n+1 (2 l − 2 n)! · , 2 k − 2 n + 1 n!( l − n)!( l − 2 n)!
−1 < x < 1, l ≥ 1. (10.1-13) l−1
(l = −
) (2 l)! . 2 l ( l!)2
(l =
)
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(10.1-6)
(2 n)!! = (2 n)(2 n − 2)(2 n − 4) · · · 6 · 4 · 2 (2 n − 1)!! = (2 n − 1)(2 n − 3)(2 n − 5) · · · 5 · 3 · 1 (2 n)! = (2 n)!!(2 n − 1)!! • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
§10.1.
[ l/2]
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k=1 [ l/2]
(−1) k (−1) k
k=1
(2 l − 2 k)(2 l − 2 k − 1) · · · ( l − 2 k + 1) l−2 k x 2 l k!( l − k)! (2 l − 2 k)! l −2 k x = P l ( x). 2 l k!( l − k)!( l − 2 k)!
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(10.1-14) x = ±1
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§10.1. Q l ( x) C2 = 0 y( x) = C1 P l ( x)
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10.1.3 Legendre
0
10.1.4 Legendre
Legendre N2 l =
+1 −1
P l ( x)
Nl
[ P l ( x)]2 d x.
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§10.1. P l ( x) Nl Nl = 2 , 2l + 1 l = 0, 1, 2, 3, · · · . (10.1-7)
=
3. Legendre Cauchy (24-3) 1 1 ( z2 − 1) l P l ( x) = d z, 2πi 2 l C ( z − x) l+1 z z=x P l ( x) = 1 π
π 0
(10.1-7) (10.1-8) Schl¨ afli
C
(10.1-8) √ l 2 x + i 1 − x cos ψ dψ.
l! ( x2) l− k ( l − k)! k!
=
k=0
(−1) k
1 2 l −2 k x . 2 l k!( l − k)! 2l − 2 k l l k ≤ l /2
l
2l − 2 k ≥ l
1 dl 2 ( x − 1) l l l 2 l! d x • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
2x dx
Legendre y1( x) (9.2-7) y0 ( x)
P l ( x) 1 dx 2 2 (1 − x ) [ P l ( x)]
e 1−x2 Q l ( x) = P l ( x) d x = P l ( x) 2 [ P l ( x)] . . . Legendre . .
(10.1-12)
Laplace (9.1-3) ∂Y 1 ∂2 Y sin θ + + l( l + 1)Y = 0. ∂θ sin2 θ ∂ϕ2 Y (θ, ϕ) . . . —— (91-37)
Y (θ, ϕ) = ( A cos mϕ + B sin mϕ)Θ(θ), Legendre 2 dΘ m2 2 d Θ (1 − x ) 2 − 2 x + l( l + 1) − Θ = 0, x = cos θ, dx dx 1 − x2 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Θ(θ)
k=0 [ l/2]
6/54 (2 l − 2 n)! , 2 n! l( l − n)!( l − 2 n)! Legendre (−1) k n = 0, 1, 2, · · · .
(10.1-3)
(2 l − 2 k)! l −2 k x , 2 k! l( l − k)!( l − 2 k)! l/2 l l ,
3/54 § 10.1 m=0
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§10.1.
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§10.1
m=0 z
2
Y (θ, ϕ) = . . . . . .
×Θ Θ (θ )
ϕ Θ ( x) Legendre
§10.1. P l (0) 2n + 1 l= P2 n+1( x) P2 n+1(0) = 0. 2n P2 n( x) (10.1-4) P l ( x) l=
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(10.1-5) k = l /2 = n
p2 n(0) = (−1) n
(2 n)! n (2 n)! = ( − 1) 2 n n!2 n n! [(2 n)!!]2 (2 n − 1)!! = (−1) n (2 n)!!
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§10.1. Q l ( x) ( Q l ( x) =
k
13/54 ) 1 1+x 1 P l ( x) ln + 2 1 − x 2l
l −1 2
x l−1−2 k
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—
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§ 9.1 Helmholtz 1 ∂ sin θ ∂θ
d2 Θ dΘ (1 − x ) 2 − 2 x + l( l + 1)Θ = 0. dx dx
(10.1-1)
10.1.1 Legendre
1. Legendre Legendre (9.2-6) (10.1-1) (9.2-7) § 9.2 (9.2-8)
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(10.1-11)
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§10.1.
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10.1.2
l P l ( x) (9.2-8) (9.3-17)
Legendre
∗
Legendre l= l= Wronski (9.3-16)
§10.1. Legendre ) (9.2-12) (9.2-7) l Φ(ϕ) = P l ( x) Legendre xl al = (2 l)! . 2 l ( l!)2 (9.2-5) ( k + 2)( k + 1) ak = a k+2, ( k − l)( k + l + 1) (9.2-8) Legendre P l ( x) Y (θ, ϕ) x = ±1 ( θ = 0, π “ x = ±1
§10.1.
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1 P3( x) = (5 x3 − 3 x) 2 1 = (5 cos 3θ + 3 cos θ) 8 1 P4( x) = (35 x4 − 30 x2 + 3) 8 1 = (35 cos 4θ + 20 cos 2θ + 9) 64 1 P5( x) = (63 x5 − 70 x3 + 15 x) 8 1 (63 cos 5θ + 35 cos 3θ + 30 cos θ) = 128 P6( x) = 1 (231 x6 − 315 x4 + 105 x2 − 5) 16 1 = (231 cos 6θ + 126 cos 4θ + 105 cos 2θ + 50) 512 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
§10.1. 2. Legendre Legendre 1 dl 2 P l ( x) = l ( x − 1) l . l 2 l! d x Rodrigues ( x2 − 1) l 1 2 1 l ( x − 1) = 2 l l! 2 l l!
∞ ∞
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ห้องสมุดไป่ตู้
(10.1-7)
(−1) k
k=0
Sturm-Livouville Legendre
+1 −1 π
(9.4-12) (−1, +1) l, k l, . ρ( x) = 1 ρ(θ) = sin θ
P k( x) P l ( x)d x = 0,
k
(10.1-15) (10.1-16)
P k(cos θ) P l (cos θ) sin θdθ = 0,
(10.1-4)
[ l/2] l = 2 l/2,
( l − 1)/2,
Legendre P0 ( x) = 1 P1( x) = x = cos θ 1 P2( x) = (3 x2 − 1) = 2 • First • Prev 1 (3 cos 2θ + 1) 4 • Next • Last • Go Back • Full Screen • Close • Quit