1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

11.2 • INVARIANCE OF LAPLACE'S EQUATION ANO THE DIRICHLET PROBLEM 427

y

¢ (x, y) =constant

• ""'"'"~

; (x, y) = K1

whenlzl = I

Figure 11.3 The harmonic function <I.> (x, y) = K 1 + K~n~Ki In lzl.

•EXAMPLE 11. 3 Find the function (x, y) that is harmonic in the annulus
1 < lzl < R and takes on the boundary values

<t> (x, y) = Ki, when lzl = 1, and

<t> (x, y) = K2, when lzl = R.

Solution This problem is a companion to the one in Example 11.2. Here we
use the fact that In lzl is a harmonic function, for all z ~ 0. The solution is

K2- K1

<t>(x,y)= K1+ lnR ln lzl,

and the level curves <t> ( x, y) = constant are concentric circles, as illustrated in

Figure 11 .3.

11.2 Invariance of Laplace's Equation and the Dirichlet Problem