2019-12.md

course	course_year	question_number	tags	title	year
Complex Methods	IB	12	IB 2019 Complex Methods	Paper 3, Section I, D	2019

By considering the transformation $w=i(1-z) /(1+z)$, find a solution to Laplace's equation $\nabla^{2} \phi=0$ inside the unit disc $D \subset \mathbb{C}$, subject to the boundary conditions

$$\left.\phi\right|{|z|=1}= \begin{cases}\phi{0} & \text { for } \arg (z) \in(0, \pi) \ -\phi_{0} & \text { for } \arg (z) \in(\pi, 2 \pi)\end{cases}$$

where $\phi_{0}$ is constant. Give your answer in terms of $(x, y)=(\operatorname{Re} z, \operatorname{Im} z)$.