&!
R.
x = r cos Ć, y = r sin Ć.
uxx + uyy = 0
1 1
vrr + vr + vĆĆ = 0.
r r2
u | "&! = h, h : "&! R,
v
v(R, Ć) = h(Ć).
r > 0
(x, y) = (0, 0) v

+
limr0 v (r, Ć) 2Ą h
h
h
a0 "
h (Ć) = + (an cos nĆ + bn sin nĆ) ,
2
n=1
Ä„ Ä„
1 1
an = h (È) cos nÈ dÈ, bn = h (È) sin È dÈ.
Ä„ Ä„
-Ä„ -Ä„
 v (r, Ć) = f (r) g (Ć)
r2f (r) rf (r) g (Ć)
+ + = 0,
f (r) f (r) g (Ć)
r2f (r) + rf (r) g (Ć)
=  = - .
f (r) g (Ć)
g 2Ä„
g (Ć) + g (Ć) = 0
2Ä„ n = n2 n = 0, 1, 2, ...
gn (Ć) = cn cos nĆ + dn sin nĆ,
cn dn n2
fn
r2fn (r) + rfn (r) - n2fn (r) = 0.
r rn r r-n n = 0
r 1 r ln r v
"
v (r, Ć) = fn (r) gn (Ć)
n=0
"
= c0 + d0 ln r + cnrn + dnr-n (cn cos nĆ + dn sin nĆ) .
n=1
v
+
limr0 v (r, Ć) , dn = 0 n e" 0
"
h (Ć) = v (R, Ć) = c0 + cnRn (cn cos nĆ + dn sin nĆ) .
n=1
h
a0
c0 = , cnRncn = an, cnRndn = bn,
2
n
a0 " r
v (r, Ć) = + (an cos nĆ + bn sin nĆ) .
2 R
n=1
B (0, R1) ‚"
B (0, R)
0 < q < 1 (an) p " N
"
annpqn
n=1
v B(0, R)
h
lim v (r, Ć) = h (Ć) .
rR-
v
an bn
cos nÈ cos nĆ + sin nÈ sin nĆ = cos n (Ć - È) ,
Ä„
"
n
1 1 r
v (r, Ć) = h (È) + cos n (Ć - È) dÈ.
Ä„ 2 R
n=1
-Ä„
r
z = ei(Ć-È).
R
" "
n
z r
zn = = (cos n (Ć - È) + i sin n (Ć - È)) ,
1 - z R
n=1 n=1
"
z
Re zn = Re ,
1 - z
n=1
1 z 1 1 + z 1 (1 + z) (1 - z)
Re + = Re = Re
2 1 - z 2 1 - z 2
|1 - z|2
2
r
1 -
1 1 - |z|2 1
R
= =
2
r
2
|1 - z|2 2 - (cos (Ć - È) + i sin (Ć - È))
1
R
1 R2 - r2
=
2 2
2 r r
R2 1 - cos (Ć - È) + sin2 (Ć - È)
R R
1 R2 - r2
= .
2 R2 - 2Rr cos (Ć - È) + r2
Ä„
1 (R2 - r2) h (È)
v (r, Ć) = dÈ
2Ä„ R2 - 2Rr cos (Ć - È) + r2
-Ä„
uxx + uyy = 0, u|"B(0,R) = h,
x = r cos Ć, y = r sin Ć, h
ReiĆ Ć.