11.2 • INVARIANCE OF LAPLACE'S EQUATION AND THE 0IR.ICHLET PROBLEM 431•EXAMPLE 11.6 Find the function
half-plane Im(z) > 0 and takes on the boundary values indicated in Figure 11.5.
Solution This is a four-value Dirichlet problem in the upper half-plane defined
by Im (z) > 0. For the z plane, the solution in Equation (11-5) becomes
1 3
k=l
Here we have <l{) = 4, a1 = 1, a2 = 3, and aa = 2 and x1 = -1, x2 = 0, and
x3 = 1, which we substitute into the equation for</> to obtain
4 - 1 1 - 3 3-2
7r 7r 7r
= 2 + ~Arctan-y-- ~Arctan~ + ~Arctan-Y-.
7r x+l 7r x 7r x - 1
91(x, 0)= 2Figure 11.5 The boundary values for the Dirichlet problem.•EXAMPLE 11.7 Find the function
half-plane Im(z) > 0 and takes on the boundary values