11.3 • POISSON'S INTEGRAL FORMULA FOR THE UPPER HALF-PLANE 441•EXAMPLE 11.11 Find the function <P(x,y) that is harmonic in the upper
half-plane Irn (z) > 0 and has the boundary values
t/>(x, 0) = 1, for -1<x<1, and t/>(x, 0) = 0, for lxl > 1.
Solution Using Equation {11-12), we obtain
A. ( ) - JL 11
dt - 2:.11
'I' x, y - 2 - ydt 2.
1r -1 (x - t) + y^2 -ir -1 (x - t) + y^2
Using the antiderivative in Equation (11-15), we write this solution as
-ir x - t t=-1
= 2:.Arctan- Y- - 2:.Arctan- Y-.
-ir x - 1 7r x + l- EXAMPLE 11.12 Find the function <P (x, y) that is harmonic in the upper
half-plane Im (z) > 0 and has the boundary values
(x, 0) = x, for -1<x<1, and (x, 0) = 0 , for lxl > 1.
Solution Using Equation (11-12), we obtain
A. ( ) y 1
1
'I' x , y = - t dt 2
7r -I (x - t) + y^2
y1
1
(x-t)(-1) dt x1
1
ydt