Алексеев Г.В. Классические методы математической физики. Часть 2
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P
Z
P (x) = (1 − x
2
)
m/2
Z( x).
x
P
′
(x) =
m
2
1 − x
2
m/2−1
(−2x)Z + (1 − x
2
)
m/2
Z
′
,
P
′′
(x) =
h
m
2
m
2
− 1
1 − x
2
m/2−2
(−2x)
2
− m
1 − x
2
m/2−1
i
Z+
+2
h
m
2
1 − x
2
m/2−1
(−2x)
i
Z
′
+ (1 − x
2
)
m/2
Z
′′
.
(1 − x
2
)P
′′
− 2xP
′
+
λ −
m
2
1 − x
2
P = 0
P, P
′
P
′′
Z
1 − x
2
m/2+1
Z
′′
+
h
−2mx(1 − x
2
)
m/2
− 2x(1 − x
2
)
m/2
i
Z
′
+
+
n
1 − x
2
m/2−1
m(m − 2)x
2
− m(1 − x
2
) + 2mx
2
+
+
λ −
m
2
1 − x
2
(1 − x
2
)
Z = 0.
(1 − x
2
)
m/2
(1 − x
2
)Z
′′
− 2x(m + 1)Z
′
+ [λ − m(m + 1)]Z = 0
d
dx
h
1 − x
2
m+1
Z
′
i
+ [λ − m(m + 1)] (1 − x
2
)
m
Z = 0.
d
dx
(1 − x
2
)
m+1
d
dx
d
m
P
n
dx
m
+ [λ − m(m + 1)](1 − x
2
)
m
d
m
P
n
dx
m
= 0,
m
d
m
P
n
/dx
m
P
n
[−1, 1] λ = λ
n
≡ n(n + 1)
n = 0, 1, 2, ...
[−1, 1] λ = λ
n
d
m
P
n
/dx
m
P
n
P (x) ≡ P
m
n
(x)
λ = λ
n
= n(n + 1)
P
m
n
(x) = (1 − x
2
)
m/2
d
m
P
n
(x)
dx
m
.
L
2
(−1, 1)
m m − 1
d
dx
(1 − x
2
)
m
d
m
P
n
(x)
dx
m
+[λ − m(m − 1) ] (1−x
2
)
m−1
d
m−1
P
n
(x)
dx
m−1
= 0.
L
m
n,k
=
1
Z
−1
P
m
n
(x)P
m
k
(x)dx =
1
Z
−1
(1 − x
2
)
m
d
m
P
n
dx
m
d
m
P
k
dx
m
dx.
L
m
n,k
=
(1 − x
2
)
m
d
m
P
n
dx
m
d
m−1
P
k
dx
m−1
1
−1
−
1
Z
−1
d
m−1
P
k
dx
m−1
d
dx
(1 − x
2
)
m
d
m
P
n
dx
m
dx =
= [n(n + 1) − m(m − 1)]
1
Z
−1
(1 − x
2
)
m−1
d
m−1
P
k
dx
m−1
d
m−1
P
n
dx
m−1
dx =
= [n(n + 1) − m(m − 1) ]L
m−1
n,k
= (n + m)(n − m + 1)L
m−1
n,k
.
L
m
n,k
= (n + m)(n − m + 1)(n + m − 1)(n − m + 2)L
m−2
n,k
= ... =
= (n + m)(n + m − 1)...(n + 1)n(n − 1)...(n − m + 1)L
0
n,k
=
=
(n + m)! (n − m)!
n!(n − m + 1)!
L
0
n,k
=
(n + m)!
(n − m)!
L
0
n,k
.
L
0
n,k
≡
1
Z
−1
P
n
(x)P
k
(x)dx =
0, k 6= n,
2
2n+1
, k = n.
1
Z
−1
P
m
n
(x)P
m
k
(x)dx =
0, k 6= n,
2
(2n+1)
(n+m)!
(n−m)!
, k = n.
kP
m
n
k
P
m
n
kP
m
n
k
2
≡
1
Z
−1
[P
m
n
(x)]
2
dx =
2
(2n + 1)
(n + m)!
(n − m)!
.
P
m
n
kP
m
n
k = 1
m n
m ≥ 0
{P
m
n
(x)}
∞
n=m
L
2
(−1, 1)
S
1
λ = λ
n
, n = 0, 1, ...
P
m
n
(cosθ), P
m
n
(cosθ)sinmϕ P
m
n
(cosθ)cosmϕ,
m = 0, 1, 2, ..., n, n = 0, 1, 2, ... .
Y
m
n
cosmϕ
sinmϕ
m = 0 Y
0
n
(θ, ϕ) = P
n
(cosθ),
m = 1 Y
−1
n
(θ, ϕ) = P
1
n
(cosθ)cosϕ, Y
1
n
(θ, ϕ) = P
1
n
(cosθ)sinϕ,
...
m = n Y
−n
n
(θ, ϕ) = P
n
n
(cosθ)cosnϕ, Y
n
n
(θ, ϕ) = P
n
n
(cosθ)sinnϕ.
Y
m
n
n
n 2n + 1
λ
n
cosmϕ sinmϕ
Y
0
n
≡ P
n
(cosθ) ϕ
P
n
n (−1, 1) S
1
n+1 Y
0
n
S
1
n m m
Y
m
n
n
Y
n
(θ, ϕ) =
n
X
m=0
(a
mn
cos mϕ + b
mn
sin mϕ)P
m
n
(cos θ),
a
mn
b
mn
Y
n
(θ, ϕ) =
n
X
m=−n
c
m
n
Y
m
n
(θ, ϕ), c
m
n
=
a
mn
,
m ≤ 0,
b
mn
,
m > 0
λ = λ
n
n
L
2
(S
1
) kY
m
n
k ≡
R
S
1
[Y
m
n
]
2
dσ
Y
m
n
2π
Z
0
cos
2
mϕdϕ =
2π
Z
0
sin
2
mϕdϕ = πε
m
, ε
m
=
2, m = 0,
1, m 6= 0,
kY
m
n
k
2
=
π
Z
0
2π
Z
0
[Y
m
n
(θ, ϕ)]
2
sin θdθdϕ =
1
Z
−1
[P
|m|
n
(x)]
2
dx
2π
Z
0
cos
2
mϕ
sin
2
mϕ
dϕ ⇒
kY
m
n
k
2
=
2πε
m
(2n + 1)
(n + |m|)!
(n − |m|)!
.
{1, cos ϕ, sin ϕ, ..., cos mϕ, sin mϕ, ...}
L
2
(0, 2π) L
2
(−1, 1)
{P
m
n
(x)}
∞
n=m
m ≥ 0
L
2
(S
1
)
{Y
m
n
(θ, ϕ), n = 0, 1, ...; m = 0, ±1, ..., ±n}.
L
2
(S
1
)
g ∈ L
2
(S
1
)
Y
m
n
g(θ, ϕ) =
∞
X
n=0
Y
n
(θ, ϕ) =
∞
X
n=0
n
X
m=−n
c
m
n
Y
m
n
(θ, ϕ),
g L
2
(S
1
) c
m
n
Y
l
k
(θ, ϕ)
S
1
{Y
l
k
}
c
m
n
=
(2n + 1)
2πε
m
(n − |m|)!
(n + |m|)!
π
Z
0
2π
Z
0
g(θ, ϕ)Y
m
n
(θ, ϕ) sin θdθdϕ.
u
n
n S
1
⊂ R
3
n
n u
n
n
∆u
n
= 0 R
3
.
2n+1
u
m
n
n m = 0, ±1, ... , ±n
u
m
n
(r, θ, ϕ) = r
n
Y
m
n
(θ, ϕ), n = 0, 1, ... , |m| ≤ n,
λ λ
n
= n(n + 1)
r
2
R
′′
+ 2rR
′
+ [k
2
r
2
− n(n + 1)]R = 0.
z = kr, w(z) = R(r)
√
z
z
2
w
′′
+ zw
′
+ (z
2
− ν
2
)w = 0, ν = n + 1/2.
ν
J
ν
(z)
N
ν
(z) H
(1)
ν
(z) H
(2)
ν
(z)
j
n
(kr) =
r
π
2kr
J
n+1/2
(kr) , y
n
(kr) =
r
π
2kr
N
n+1/2
(kr) ,
h
(1)
n
(kr) =
r
π
2kr
H
(1)
n+1/2
(kr) , h
(2)
n
(kr) =
r
π
2kr
H
(2)
n+1/2
(kr) ,
p
π/2
ν = n + 1/2
h
(1)
n
h
(2)
n
j
n
y
n
h
(1)
n
(kr) = j
n
(kr) + iy
n
(kr) ,
h
(2)
n
(kr) = j
n
(kr) − iy
n
(kr) ,
j
n
h
(1)
n
(kr)
h
(2)
n
(kr)
h
(1)
n
(kr)
h
(2)
n
(kr)
u
n
(r, θ, ϕ) = h
(1)
n
(kr) Y
n
(θ, ϕ) = h
(1)
n
(kr)
n
X
m=−n
a
m
n
Y
m
n
(θ, ϕ),
a
m
n
n
n
n = 0
n = 1
n = 2
∞
X
n=0
h
(1)
n
(kr) Y
n
(θ, ϕ) ≡
∞
X
n=0
h
(1)
n
(kr)
n
X
m=−n
a
m
n
Y
m
n
(θ, ϕ),
a
m
n
r, θ ϕ
S
a
m
n
α = co nst ≥ 0, β = const ≥ 0 α + β 6= 0.
g
Y
m
n
c
m
n
∂u/∂n = ∂u/∂r
r = a
∂u
∂r
= k
∞
X
n=0
(h
(1)
n
)
′
(kr)
n
X
m=−n
a
m
n
Y
m
n
(θ, ϕ),
αu + β
∂u
∂n
=
∞
X
n=0
h
αh
(1)
n
(ka) + kβh
(1)
′
n
(ka)
i
n
X
m=−n
a
m
n
Y
m
n
(θ, ϕ).
a
m
n
c
m
n
g
a
m
n
=
c
m
n
αh
(1)
n
(ka) + kβh
(1)
′
n
(ka)
.
a
m
n
g
g ∈ C
3
(S)
D S
D g ∈ C
3
(S)
∂u
∂t
= a
2
∆u + f, (1)
a
2
= const > 0 f
Ω R
3
Γ Q
T
= Ω ×( 0, T ] Σ
T
= Γ ×(0, T ], Ω
0
= Ω ×{t = 0}
0 < T < ∞ Q
T
R
4
= R
3
x
×R
1
t
Σ
T
Ω
0
Q
T
Q
T
∂u
∂t
= a
2
∆u ≡ a
2
∂
2
u
∂x
2
+
∂
2
u
∂y
2
+
∂
2
u
∂z
2
,
u|
Γ
= g(x, t), (x, t) ∈ Γ × (0, T ]
u|
t=0
= ϕ(x), x ∈ Ω.
x = (x, y, z) u
Ω a
2
> 0
g ϕ
Γ
Ω
Ω
C
2,1
(Q
T
) C(Q
T
)
Q
T
t C(Q
T
)
C(Q
T
) Q
T
= Ω × [0, T ]
Q
T
u ∈ C
2,1
(Q
T
) ∩ C(Q
T
)
Q
T
Γ × (0, T ] Q
T
Ω × {t = 0}
u
(x, t) ∈ Q
T
(x, t) −u
M = max
(x,t)∈
Q
T
u(x, t), m = max
(x,t)∈
Σ
T
∪Ω
0
u(x, t).
M ≥ m M = m
u ∈ C
2,1
(Q
T
) ∩C(Q
T
)
u
M > m Q
T
u
(x
0
, t
0
) = (x
0
, y
0
, z
0
, t
0
) ∈ Ω × (0, T ]