Untformly Valtd Soluttons of Laplace Equatton 
Using the previous result (15), it seems plausible to try an a priori 
expansion under the form : 
pit cree 1 0 ep +. (47) 
It will be seen, later on, that expansion ( KZ) as incomplete, because 
the asymptotic sequence must contain logarithmic terms, 
~(1) 
Let us consider the problem to be solved in order to find#¢, 
b)"! ( z,%,) [a a) + alll Z| +i [ By) Bz »| 
where Au A Y ) and Bit Y ) are real functions on the positive part of 
the real axis. The holomorphic functions AN z Z )and B!'\ Zz ) are obtai- 
ned after solving the Hilbert problems deduced from the boundary 
conditions 
(1) 
361 (at. 2) (et 22K) Pee od 
2 SES =O eee 
0Z OZ OZ ioe Toe 
T dee xed aor 
Be i Vedat Ube Ee at SA Bi tty . sats) 
cay ee Ar) nel Be ae thy a so §) 
p V2 
A UA imps 4668 dolasthe/ PY (18) 
pile i Le a Alek § {P eas /p (19) 
The following notations are used : 
+ 
2 OU Sani se REL pile: Ona 
The solution of equation- (18) is : al’ = Deicke + homogeneous 
eee of equation (19) is: B" = __bolP)(s)_ z5-log Z + homo- 
geneous solution. 
a Jo 2 to the previous variable z = nz =Y+iZ wean see that 
a OS e)(s) Z—- 5. does not depend upon 7. But the func- 
beth 1 
=== 1 
- 7 . 1% ie 50!P)(s) ices t i so”) oe 
ne hon Moe eee afi 
1749 
