11.3 • POISSON'S INTEGRAL FOR.MULA FOR THE UPPER HALF-PLANE 439- Find the function <fi (x, y) that is harmonic in the quarter-disk x > 0, y > 0, lzl < 1
and has the boundary values
<P(x, y) = 3,
<P (x, 0) = -3,
<P (0, y) = -3,forx+iy= z=e'^9 , 0<11<~;
for 0 ~ x < l;
forO<y<l.- Find the function t/> (x, y) that is harmonic in the unit disk lzl < 1 and has the
boundary values
t/>(x, y) = 1,tf>(x, y) = 0,for x + iy = z = e•^6 ,
for x + iy = z = e^19 ,
11.3 Poisson's Integral Formula for the Upper Half-Plane
The Dirichlet problem for the upper half-plane Im(z) > 0 is to find a function
(x, y) that is harmonic in the upper half-plane and has the boundary values