11.2 • INVARIANCE OF LAPLACE'S EQUATION AND THE· DIRICHLET PROBLEM 429Substituting these equations into the formula for <I> ( u, v) and using Equation
v(x,y) 2x
(11-2), we find that <fi(x,y) = Arctan- ( - ) = Arctan 2 2 is harmonicu x,y x +y -1
for lzl < L
Let D be a domain whose boundary is made up of piecewise smooth contours
joined end to end. The Dirichlet problem is to find a function ¢> that is
harmonic in D such that ¢> takes on prescribed values at points on the boundary.
Let's first look at this problem in the upper half-plane.
- EXAMPLE 11.5 Show that the function
1 v 1(u, v) = -Arctan--= -Arg (w -u<>)
7r u - uo 7r(11-4)L~ harmonic in the upper half-plane Im(w) > 0 and takes on the boundary values
( u, 0) = 0 for u > uo and
( u, O) = 1 for u < u 0 •
Solution The function1 1 ig (w) = -Log(w -uo) = -ln lw -uol + -Arg (w -u<>)
7r 7r 7ris analytic in the upper half-plane Im (w) > 0, and its imaginary part is the
harmonic function ~Arg (w -u<>)·Remark 11.1 We let t be a real number and use the convention Arctan ( ±oo) =
~ so that the function Arctan t denotes the branch of the inverse tangent that
lies in the range 0 < Arctan t < 7r. Doing so permits us to write the solution in
Equation (11-4) as <I> (u, v) = ~Arctan (-v-). •
7r U- 'UQ