Interchange the x and y axes in Sec. 139 to write the

Chapter 0, Problem 12.21

(choose chapter or problem)

Interchange the x and y axes in Sec. 139 to write the solution I !'" x FU) lJ(x.y) = - . ~ .:i. dr (X > 0) ;r . - '- (f - ' ) ' .\ of the Dirichlet problem for the half plane x > 0. Then wri tc F(r) = {I when l.rl 1. and obtain these expressions for U and its harmonic conjugate -V: I( r+l r-1) U(x. y) = - arctan - -- - arctan - -- . ;r x x where -;r /2 .::: arctan r .::: ;r /2. Also. show that 1 I x 2 +(r+1) 2 \ 1 (.\. \') = - In , . , . 2;r x- ~ (y - 1 )- \l(.r.y) + iU