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U(ρ, θ, ϕ) = U
0
"
3
2
ρ
l
cos θ −
7
8
µ
ρ
l
¶
3
5 cos
3
θ − 3 cos θ
2
+
+
11
6
µ
ρ
l
¶
5
63 cos
5
θ − 70 cos
3
θ + 15 cos θ
8
+ ...
#
.
∞
P
m=0
a
m
ρ
m
P
m
(cos θ)
a
m
= 0 m a
m
=
(2m+1)U
0
l
m
1
R
0
P
m
(τ)dτ
m
U
(
ρ, θ, ϕ
) =
U
0
2
3
−
U
0
µ
ρ
l
¶
2
3 cos
2
θ − 1
3
.
ϕ(r, θ) =
∞
X
n=0
µ
1
r
¶
n
P
n
(cos θ); (r > 1).
f(x) =
1
2
+
3
2
2
P
1
(x) −
7 · 2!
2
4
· 2! · 1!
P
3
(x) +
1! · 4!
2
6
· 3! · 2!
P
5
(x) − ...
U(r, θ) =
∞
P
n=0
A
n
³
r
R
´
n
· P
n
(cos θ), (r < R);
∞
P
n=0
A
n
³
R
r
´
n+1
· P
n
(cos θ), (r > R),
A
n
=
2n + 1
2
π
Z
0
f(θ)P
n
(cos θ) sin θdθ.
u(r, θ) =
1
3
(1 − r
2
) + r
2
cos
2
θ.
u(r, θ, ϕ) =
8
15
µ
r
R
¶
3
P
1
3
(cos θ) −
1
5
r
R
P
1
1
(cos θ) cos ϕ.
ÎÒÂÅÒÛ È ÓÊÀÇÀÍÈß.
132.
" µ ¶3
3ρ 7 ρ 5 cos3 θ − 3 cos θ
U (ρ, θ, ϕ) = U0 cos θ − +
2l 8 l 2
µ ¶5 #
11 ρ 63 cos5 θ − 70 cos3 θ + 15 cos θ
+ + ... .
6 l 8
∞
P
Ó ê à ç à í è å. Ðåøåíèå èùåòñÿ âèäå ðÿäà am ρm Pm (cos θ)
m=0
(2m+1)U0 R1
ãäå am = 0 äëÿ âñåõ ÷åòíûõ m , è am = Pm (τ )dτ äëÿ
lm
0
íå÷åòíûõ m.
133. µ ¶2
2 ρ 3 cos2 θ − 1
U (ρ, θ, ϕ) = U0 − U0 .
3 l 3
134. ∞ µ ¶n
X 1
ϕ(r, θ) = Pn (cos θ); (r > 1).
n=0 r
1 3 7 · 2! 1! · 4!
f (x) = + 2 P1 (x) − 4 P3 (x) + 6 P5 (x) − ...
2 2 2 · 2! · 1! 2 · 3! · 2!
135.
∞ ³ ´n
P
An Rr · Pn (cos θ), (r < R);
U (r, θ) = n=0
∞
P ³ ´n+1
An Rr · Pn (cos θ), åñëè (r > R),
n=0
ãäå
π
2n + 1 Z
An = f (θ)Pn (cos θ) sin θdθ.
2
0
 ÷àñòíîì ñëó÷àå u(r, θ) = 13 (1
− r2 ) + r2 cos2 θ.
136.
µ ¶
8 r 3 1 1r 1
u(r, θ, ϕ) = P3 (cos θ) − P (cos θ) cos ϕ.
15 R 5R 1
75
