0.8
0.4
0.0--0.4
11.3 • POISSON'S INTEGRAL FORMULA FOR THE UPPER HALF-PLANE 443-4Figure 11 .15 The graph of ti = if> (x, y) with the boundary values if> (x, 0) = x, for
lxl < 1, ef:>(x,O) = -1, for x < -1, and ef:>(x,O) = 1, for x > 1.
-------~EXERCISES FOR SECTION 11.3
- Use Poisson's integral formula t o find the harmonic function if> (x, y) in the upper
half-plane that takes on the boundary values
ef:>(t,0) = u (t) = 0,
ef:>(t,0) = u (t) = t,
ef:>(t,O) = U (t) = 0,fort< O;
for 0 < t < 1;
for 1 < t.- Use Poisson's integral formula to find t he harmonic function if> (x, y) in the upper
half-plane that takes on the boundary values
ef:>(t,0) = u (t) = 0,
ef:>(t,O) = U (t) = t,
ef:>(t,O) = U(t) =l,fort< O;
for 0 < t < 1;
forl<t.- Use Poisson's integral formula for the upper half-p lane to conclude that
ef>(x, y) = e -Ycosx = -y j"° cost dt.
Tr -oo (x -t)^2 + y^2- Use Poisson's integral formula for the upper half-plane to conclude t hat
-.( 'I' x, y ) = e -v Slll. X = -y j"° ( sint )2 dt
Tr -oo x-t + y^2 ·