Untformly Valtd Solutions of Laplace Equatton 
nm + + 
2 # (2) 5 = 
ph \\ l il 
3 Bil : P + pore wks 
Sat (alvie o* (ait Pe 
| 
= {2} 
ek 2 og om es 1 = We jE. ce 
= Logo. oe H, ©, | ad ooh ake Yaar oe 
Y Y ! 1 
n EOME AVE le (tf 
+ Log n |: le FH @, +H, 2 51t 
0 
+ dela [le iP ft Veo he Oe ae ye 
n 
+ a | Hy 2 ca 5 A eR = oe a 5 es: 
lyl 
ae 
Za Y qo 
gles, yj anetine vay ant, abe ee ee ay Bao) 
0 Viasl 1 3 2 0 2 
x (p) z (p) , (p) t (e) 
is 60 ne 8 ) 6 
P er 238 0 i) 1 
0 = iat ets P, = - d,+ = +-—- Log IYI, eS = = P 
u (e) o (e) 
1, = - 1 Log | ¥| + 2 - ot 
= (p) bs (p) ~_ d \"7 (e) 
pll- =f $9 gir = oP rages ce = - b4 
0 Pp eiter ae. 1 + a Log | Y| + 2 ’ 2 = 
The expanded solution contains some terms o (Log n)?and 
fe) ( n4 (Log 1)?) because the Hilbert problem : 
oe See e 6 = - 7 p= ly 
has the particular solution uw (Z) = =e ( Log z bs 
After analytical calculations, the result is found to be : 
2 2 ~ (2) ~(2) ~(2 2 = ~ 
$e) (Log n) ®9 + Log 7 rs oP), n%h(Log n) gl?) , nh Log noe ert (2a) 
Bt pe i ere en. BR) rate ag 
202 2 “aed. 4alkty G2 2 
> 
Lot 
