2006-35.md

course	course_year	question_number	tags	title	year
Complex Methods	IB	35	IB 2006 Complex Methods	3.I.5D	2006

The transformation

$$w=i\left(\frac{1-z}{1+z}\right)$$

maps conformally the interior of the unit disc $D$ onto the upper half-plane $H_{+}$, and maps the upper and lower unit semicircles $C_{+}$and $C_{-}$onto the positive and negative real axis $\mathbb{R}{+}$and $\mathbb{R}{-}$, respectively.

Consider the Dirichlet problem in the upper half-plane:

$$\frac{\partial^{2} f}{\partial u^{2}}+\frac{\partial^{2} f}{\partial v^{2}}=0 \quad \text { in } \quad H_{+} ; \quad f(u, v)= \begin{cases}1 & \text { on } \mathbb{R}{+} \ 0 & \text { on } \mathbb{R}{-}\end{cases}$$

Its solution is given by the formula

$$f(u, v)=\frac{1}{2}+\frac{1}{\pi} \arctan \left(\frac{u}{v}\right) .$$

Using this result, determine the solution to the Dirichlet problem in the unit disc:

$$\frac{\partial^{2} F}{\partial x^{2}}+\frac{\partial^{2} F}{\partial y^{2}}=0 \quad \text { in } \quad D ; \quad F(x, y)= \begin{cases}1 & \text { on } C_{+} \ 0 & \text { on } C_{-}\end{cases}$$

Briefly explain your answer.