1 1.2 • INVARIANCE OF LAPLACE'S EQUATION AND THE D IRICHLET PROBLEM 437
Figure 11.12 The graph
2 l- (x^2 +y^2 )^2 2 l-r^4
u - -Arcta.n
4
- -Arctan
4 2
@ • e·
1r xy 1r r cos sm
-------.. EXERCISES FOR SECTION 11.2For each exercise, find a solution ¢ ( x, y) of the Dirichlet problem in the domain
indicated that takes on the prescribed boundary values.
1. Find the function ef> (x, y) that is harmonic in the horizontal strip 1 < Im (z) < 2
a.nd ha.s the boundary valuesef>(x, 1) = 6, for all x, and ef>(x, 2) = -3, for all x.
- Find the function ef> (x, y) that is harmonic in the sector 0 < Argz < j a.nd ha.s
the boundary values
ef>(x, y) = 2,1r
for Argz =
3
, a.nd ef>(x, O) = 1, for x > O.- Find the function ef> (x, y) that is harmonic in the annulus 1 < lzl < 2 and has the
boundary values
ef>(x, y) = 5, when lzl = 1, and ef>(x, y) = 8, when lzl = 2.